Month 20xx Publication series of the Government’s analysis, assessment and research activities xx/20xx Arial Regular 18/20 – Sitamet elatvelmagna Otsikko Arial Bold 22/27 pt gemena – quateoffictatenfug loremipsumdolo sit dolores sitamet The Finnish Potential Output: Measurement and Medium-term Prospects Sami Jysmä, Ilkka Kiema, Tero Kuusi, Markku Lehmus January 2019 Publication series of the Government’s analysis, assessment and research activities 13/2019 DESCRIPTION Publisher and release date Prime Minister’s Office, 28.1.2019 Authors Sami Jysmä, Ilkka Kiema, Tero Kuusi, Markku Lehmus Title of publication The Finnish Potential Output: Measurement and Medium-term Prospects Name of series and number of publication Publications of the Government´s analysis, assessment and research activities 13/2019 Keywords Potential output, Output gap, Production function, Productivity Other parts of publication/ other produced versions Release date 2019 Pages 91 Language English Abstract In this report, we discuss the measurement of the potential output of the Finnish economy and the potential’s medium-term growth prospects. We apply novel approaches to the estimation of Finland’s production function (constant elasticity of substitution, CES) and the filtration of the potential output (Sequential Monte Carlo method, SMC) with the aim of improving the European Commission (EC) production function methodology in mind. Our results suggest that the cyclical component in the fluctuation of Finland’s GDP may have been historically larger than the European Commission method would suggest. We present statistical evidence which supports the replacement of the Cobb-Douglas production function with the more general CES function. The latter function leads to a distinction between labor- and capital-augmenting productivity which matters for the measurement of the potential output: the capital-augmenting productivity tends to develop in a more procyclical manner than the labor augmenting productivity, and the SMC estimation shows that there is negative covariation in the cyclical components. We further apply the SMC method to estimate the NAWRU and the labor force participation rate as well as their covariation. The results are similar with the EC estimates during the last years, but smoother during the Finnish Great Depression. We find that especially the estimates of NAWRU and the potential labor-augmenting productivity are very sensitive to the real-time uncertainty. We study the medium-term growth potential of the Finnish Economy in the years 2019–2023 using Etla’s multi- sector growth model. We find that the average GDP growth rate produced by the model forecast is 1.5% under the expected trends in technology, demography, and trade. The growth is predominately determined by information technology and the external trade. This publication is part of the implementation of the Government Plan for Analysis, Assessment and Research for 2017 (tietokayttoon.fi/en). The content is the responsibility of the producers of the information and does not necessarily represent the view of the Government. KUVAILULEHTI Julkaisija ja julkaisuaika Valtioneuvoston kanslia, 28.1.2019 Tekijät Sami Jysmä, Ilkka Kiema, Tero Kuusi, Markku Lehmus Julkaisun nimi Suomen potentiaalinen tuotanto: mittaaminen ja keski-pitkän aikavälin odotukset Julkaisusarjan nimi ja numero Valtioneuvoston selvitys- ja tutkimustoiminnan julkaisusarja 13/2019 Asiasanat Potentiaalinen tuotanto, Tuotantokuilu, Tuotantofunktio, Tuottavuus Julkaisun osat/ muut tuotetut versiot Julkaisuaika 2019 Sivuja 91 Kieli Englanti Tiivistelmä Tämä tutkimus koskee Suomen kansantalouden potentiaalisen tuotannon mittaamistapaa ja potentiaalin kasvunäky- miä keskipitkällä aikavälillä. Tutkimuksen lähtökohtana on Euroopan komission käyttämä tuotantofunktiomenetelmä. Siinä kansantalouden tuotantotapaa kuvataan erilaisten tuotantopanosten ja käyttöteknologioiden yhdistelmänä eli tuotantofunktiona. Eri komponentteihin liittyvät suhdannevaikutukset huomioidaan erikseen kokonaissuhdannevaiku- tuksen laskemiseksi. Hankkeessa komission menetelmää kehitetään mallintamalla tuotanto aikaisempaa yksityiskohtaisemmin ns. vakioi- sen työvoima- ja pääomapanoksen välisen substituutiojouston (CES) tuotantofunktion avulla. Lisäksi tuotantofunkti- on eri komponenttien suhdanneluonteista vaihtelua arvioidaan uudella Sequential Monte Carlo (SMC) -menetelmällä. Tutkimuksessa havaitaan, että CES-tuotantofunktio soveltuu paremmin kuvaamaan Suomen kansantalouden tuotan- totapaa kuin komission käyttämä yksinkertaisempi ns. Cobb-Douglas -tuotantofunktio. CES-tuotantofunktio mahdol- listaa myös eri tuotantopanosten, eli työvoiman ja pääoman, käytön tehokkuuden erillisen arvioinnin. Hankkeessa saatujen uusien tulosten mukaan laskusuhdanne alentaa pääoman käytön tehokkuutta, kun taas työvoiman käytön tehokkuuteen se vaikuttaa vain vähän ja jopa tehostavasti. Mallinnamme SMC-menetelmällä, paitsi tuottavuussarjojen potentiaalin, myös kansantalouden inflaationeutraalin tasapainotyöttömyyden eli NAWRU:n, sekä työvoiman osallistumisasteen rakenteellisen tason. Viimeaikaista kriisiä koskevat arviomme syklisistä vaikutuksista ovat samansuuntaisia komission aikaisempien arvioiden kanssa, mutta 1990-luvun lamaa koskevat arviot ovat vähemmän myötäsyklisiä. Kaiken kaikkiaan tuloksiemme mukaan suhdannevaihtelut ovat vaikuttaneet kansantalouden tuotantomäärään jonkin verran enemmän kuin komission menetelmää noudattaen on aikaisemmin arvioitu. Erityisesti NAWRU:n ja työvoiman tuottavuuden reaaliaikainen arviointi on kuitenkin uusillakin menetelmillä vaikeaa ja arviot ovat herkkiä revisioitumaan. Arvioimme myös kansantalouden potentiaalisen tuotannon kasvunäkymiä vuosina 2019–2023 Etlan sektoritasoisen kasvumallin avulla. Käyttämässämme mallissa teknologisen kehityksen, demografian ja kaupan pitkän aikavälin tren- dien vaikutuksia arvioidaan yhtenäisessä mallikehikossa. Mallin perusuralla tuotannon volyymikasvu tulee olemaan noin 1,5 % vuodessa seuraavien viiden vuoden aikana. Kasvuun vaikuttavat erityisen voimakkaasti informaatiotekno- logia ja kansainvälisestä kaupasta syntyvä kasvuvaikutus. Tämä julkaisu on toteutettu osana valtioneuvoston vuoden 2017 selvitys- ja tutkimussuunnitelman toimeenpanoa (tietokayttoon.fi). Julkaisun sisällöstä vastaavat tiedon tuottajat, eikä tekstisisältö välttämättä edusta valtioneuvoston näkemystä. PRESENTATIONSBLAD Utgivare & utgivningsdatum Statsrådets kansli, 28.1.2019 Författare Sami Jysmä, Ilkka Kiema, Tero Kuusi, Markku Lehmus Publikationens namn Den potentiella finländska produktionen: beräkning och tillväxtutsikter på medellång sikt Publikationsseriens namn och nummer Publikationsserie för statsrådets utrednings- och forskningsverksamhet 13/2019 Nyckelord Potentiell produktion, Produktionsgap, Produktionsfunktion, Produktivitet Publikationens delar /andra producerade versioner Utgivningsdatum 2019 Sidantal 91 Språk Engelska Sammandrag I denna rapport analyserar vi mätningen av den finländska potentiella produktionen och potentialens tillväxtutsikter på medellång sikt. Vi tillämpar nya metoder för att beräkna den finländska produktionsfunktionen (konstant elasticitet för substitution, CES) och filtrering av potentiell produktion (sekventiell Monte Carlo-metoden, SMC) i syfte att förbättra Europeiska kommissionens (EG) metod för att beräkna produktionsfunktion. Våra resultat tyder på att den cykliska faktorn i den finländska BNP-fluktuationen historiskt sett kan ha varit större än vad Europeiska kommissionens metod antyder. Vi presenterar statistisk bevisning som stöder användningen av mer allmän, CES produktion funktion i stället för Cobb-Douglas. Den uppkomna distinktionen mellan produktivitet som förstärker arbetskraft och kapital har betydelse för mätning av den potentiella produktionen: Produktivitet som förstärker kapital tenderar att utvecklas på ett mer procykliskt sätt än produktivitet som förstärker arbetskraft, och SMC uppskattning visar att det finns en negativ variation i de cykliska komponenterna. Vi tillämpar vidare SMC-metoden för att uppskatta NAWRU och arbetskraftsdeltagandet och deras samvariation. Potentialen liknar EG-skattningarna under de senaste åren, men smidigare under den finska stora depressionen. Vi finner att särskilt uppskattningar av NAWRU och den potentiella arbets-höjande produktiviteten är mycket känsliga för realtid osäkerhet. Vi studerar den finländska ekonomins tillväxtpotential på medellång sikt under åren 2019–2023 genom att använda ETLA:s tillväxtmodell för flera sektorer. Vi finner att den genomsnittliga BNP-tillväxten som produceras av modellprognosen är 1,5% under de förväntade trenderna inom teknik, demografi och handel. Tillväxten bestäms huvudsakligen av informationsteknologin och utrikeshandeln. Den här publikation är en del i genomförandet av statsrådets utrednings- och forskningsplan för 2017 (tietokayttoon.fi/sv). De som producerar informationen ansvarar för innehållet i publikationen. Textinnehållet återspeglar inte nödvändigtvis statsrådets ståndpunkt. Tutkimuksen suomenkielinen tiivistelmä . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Aims of this project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 CES aggregate production function and the output gap . . . . . . . . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Cobb-Douglas and the CES production function . . . . . . . . . . . . . . . . . . . . 10 2.3 An estimation procedure for the normalized CES function . . . . . . . . . . . . . . . . 12 2.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Choosing the model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Heterogeneity of productivity within the input series . . . . . . . . . . . . . . . . . . . . . 19 2.6.1 Constructing measures of capital input . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6.2 Constructing measures of labor input . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6.3 Comparing input series combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 The trend and cycle components of the productivity series . . . . . . . . . . . . . . . 23 2.8 Comparing output gaps in the Cobb-Douglas and CES specifications . . . . . . . 26 3 Estimation of the potential output by using the Sequential Monte Carlo . . . . 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The current assessment of the potential and its validity criteria . . . . . . . . . . . . 30 3.3 Estimation of the labor force variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 NAWRU and the measurement of the unemployment gap with the Phillips curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Taking stock of the recent NAWRU critique . . . . . . . . . . . . . . . . . . . . . . 34 3.3.3 Introduction of the labor force participation rate to the model . . . . . . . . . 36 3.3.4 Specification of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Total-factor productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.1 Discussion of the measurement of the TFP potential . . . . . . . . . . . . . . . 39 3.4.2 Specification of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Estimates of the NAWRU and the structural level of the participation rate . . . . 43 3.5.1 A closer look on the estimation of the model . . . . . . . . . . . . . . . . . . . . . 47 3.5.2 Alternative specifications of the Phillips curve . . . . . . . . . . . . . . . . . . . . 50 CONTENTS 3.6 Estimates of the labor- and capital-augmenting factor productivity . . . . . . . . . 52 3.6.1 A closer look on the estimated parameters . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.2 The role of other business cycle indicators in the identification . . . . . . . 57 3.7 The output gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.8 Concluding remarks on the estimation of the potential output and the output gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 Projections of the medium-term growth potential for Finland . . . . . . . . . . . . . 60 4.1 The determinants of the potential output growth in the model . . . . . . . . . . . . . 62 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Understanding the growth potential requires alternative scenarios . . . . . . . . . 68 4.4 Concluding remarks on the medium-term production potential . . . . . . . . . . . . 71 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Appendix to Section 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix to Section 3: Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix to Section 3: Smoothed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Appendix to section 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1 TUTKIMUKSEN SUOMENKIELINEN TIIVISTELMÄ Tämä tutkimus koskee Suomen kansantalouden potentiaalisen eli suhdannevaihteluista riippumattoman tuotantomäärän mittaamistapaa ja potentiaalin kasvunäkymiä keskipitkällä aikavälillä. Potentiaalin laskemisessa tutkimuksen lähtökohtana on Euroopan komission käyttämä tuotantofunktiomenetelmä. Siinä kansantalouden tuotantotapaa kuvataan erilais- ten tuotantopanosten ja käyttöteknologioiden yhdistelmänä eli tuotantofunktiona. Menetel- mässä määritellään ensin tuotantofunktio ja sen jälkeen kokonaistuotannon suhdanneluon- teinen vaihtelu lasketaan perustuen tuotantofunktion eri komponenttien vaihteluun. Hankkeessa komission tuotantofunktiomenetelmää kehitetään useilla tavoilla. Tuotantofunk- tio mallinnetaan aikaisempaa tarkemmin ns. vakioisen substituutiojouston (CES) tuotanto- funktion avulla. Käytetty CES-tuotantofunktio poikkeaa komission aikaisemmin käyttämästä ns. Cobb-Douglas -tuotantofunktiosta, koska CES-funktion käyttäytymisen kannalta kes- keinen (vakioinen) työvoima- ja pääomapanoksen välinen substituutiojousto estimoidaan aineistosta, mutta Cobb-Douglas -funktiota käytettäessä sen arvo perustuu funktion mää- ritelmään. Lisäksi hankkeessa tuotantofunktion eri komponenttien suhdanneluonteista vaihtelua arvioidaan uudella menetelmällä (ns. Sequential Monte Carlo (SMC) -menetelmä). Uusien tarkastelujen avulla saadaan entistä yksityiskohtaisempaa tietoa suhdanteiden vai- kutuksesta Suomen kansantalouteen ja toisaalta suhdanteista riippumattoman potentiaalin kehityksestä. Syitä tulosten tarkentumiseen on useita. Ensinnäkin tutkimuksessa havaitaan, että CES-tuo- tantofunktio sopii tilastollisesti paremmin kuvaamaan Suomen kansantalouden tuotantota- paa kuin komission käyttämä Cobb-Douglas -tuotantofunktio. Lisäksi CES-tuotantofunktio mahdollistaa eri tuotantopanosten, erityisesti työvoiman ja pääoman, käytön tehokkuuden erillisen arvioinnin. Hankkeessa saatujen uusien tulosten mukaan laskusuhdanne alentaa pääoman käytön tehokkuutta, kun taas työvoiman käytön tehokkuuteen se vaikuttaa vä- hemmän. Tämä tulos auttaa paremmin ymmärtämään Suomen suhdanteisiin voimakkaasti vaikuttavaa, mutta huonosti tunnettua kokonaistuottavuuden heilahtelua, ja yhdistää sen aikaisempaa enemmän pääoman kapasiteetin käytön vaihteluihin. Suhdannearvioiden täsmentymiseen vaikuttaa lisäksi myös se, että mallinnamme uudel- la SMC-menetelmällä tuotantofunktion eri komponenttien suhdannekäyttäytymistä. Uutta tarkastelussa ovat erityisesti eri komponenttien suhdannekäyttäytymisen yhteistarkastelut: Tutkimme CES-tuotantofunktiosta saatujen tuotantopanosten käytön tehokkuuksien ja työ- voimapanoksen eri komponenttien yhteisliikettä suhdanteen aikana. Havaitsemme, että pääomapanoksen käytön suhdanneluonteinen tehokkuuslasku on yhteydessä työpanoksen käytön tehokkuusnousuun. Havainto tukee aikaisempia arvioita työpanoksen rakenteen muuttumisesta suhdanteiden mukana. Työpanoksen potentiaalin arvioimiseksi hyödynnämme mallia, jossa inflaationeutraali tasapainotyöttömyys, NAWRU, arvioidaan yhtäaikaisesti työvoiman osallistumisasteen rakenteellisen tason kanssa. Ha- vaitsemme, että työttömyyden ja osallistumisasteen välillä on yhteinen suhdanneluonteinen komponentti: työttömyyden suhdanneluonteinen kasvu johtaa osallistumisasteen suhdanne- luonteiseen laskuun. Muuttujien yhtäaikaisen muutoksen huomiointi voi edesauttaa esimer- kiksi inflaation ja suhdanteiden välisen yhteyden parempaa hyödyntämistä suhdannetilan- teen arvioinnissa. Tuloksiemme mukaan suhdannevaihtelut ovat vaikuttaneet kansantalouden tuotantomää- rään jonkin verran enemmän kuin komission menetelmää noudattaen on aikaisemmin arvioitu. Tuotantofunktion eri komponenttien suhdanneluonteisen vaihtelun osalta myös 2 uusin menetelmin tuotetut viimeaikaista talouskriisiä koskevat arviot ovat tosin samansuun- taisia komission aikaisempien arvioiden kanssa. Erot ovat sen sijaan suurempia 1990-luvun laman osalta. Uudet potentiaalin arviot riippuvat huomattavasti vähemmän suhdanteista, ja siten uusi menetelmä näyttäisi vähentävän potentiaaliarvioihin liittyvää osittain ongelmallista myötäsyklisyyttä. Toisaalta erityisesti NAWRU:n ja työpanoksen käytön tehokkuuden reaa- liaikainen arviointi on uusienkin menetelmien avulla vaikeaa ja arviot ovat herkkiä revisioitu- maan. Osana tutkimusta arvioimme myös kansantalouden potentiaalisen tuotannon kasvunäkymiä vuosille 2019–2023. Hyödynsimme arvioinnissa Etlan sektoritasoista kasvumallia, jossa potentiaalisen tuotannon kasvua ajavien fundamentaalien (tuottavuuskehitys, demografia ja kaupan pitkän aikavälin rakenne) vaikutuksia potentiaaliin arvioidaan yhtenäisessä kehikos- sa. Investointi ja kulutuskäyttäytymisen oletetaan olevan kuluttajan hyvinvointia optimoivaa ja resurssien olevan tehokkaassa käytössä. Arvion perusteella tuotannon volyymikasvu tulee olemaan noin 1,5 % vuodessa seuraavien viiden vuoden aikana, tosin arvoihin liittyy huomattava määrä epävarmuutta. Kasvuun vaikuttavat erityisen voimakkaasti informaatio- teknologia ja kansainvälisestä kaupasta syntyvä kasvuvaikutus. 2 3 1 INTRODUCTION The proper timing and scaling of fiscal policy require detailed information about the cyclical situation of the economy and its longer-term production potential. The objective of fiscal poli- cy is on one hand to respond to changes in the economic cycles: to offset the loss of private economic activity in the downturn and to create new stimulus reserves and help to avoid overheating during the upturns. On the other hand, fiscal policy must also ensure fiscal sus- tainability, which depends on the development of the production potential of the economy in the longer term. However, the cyclical nature of various economic shocks and their impact on the long-term production potential is difficult to observe in practice. Potential output, typically defined as the level of output that an economy can produce without triggering above-normal inflation, is an economic concept that has no direct, observable counterpart in the data. The estimation of the potential output and the output gap (i.e. the difference between actual and potential economic activity) requires assessments on several quantities that are difficult to measure. In practice, the potential output is typically identified via the estimation of the low frequency movement of the actual production. The inflation-neutral growth of the economy is assessed by forecasting the low-frequency trend of the economy and then comparing actual economic development to this trend. Due to both genuine uncertainty and the difficulty of model choice, the potential output estimates are uncertain and often change considerably over time. This uncertainty is costly because it undermines the possibility of a counter-cyclical fiscal policy. It is especially prob- lematic in the European policy context because the statistical estimates of the production potential and the output gap have been given a central role in the EU’s fiscal framework (European Commission 2018A). The measurements are used in calculation of one of the most central fiscal policy indicators of the framework: the structural budget balance (see, for example, Mourre et al., 2013; Havik et al., 2014)1. Uncertainty will never be completely eliminated, but it is nevertheless an important goal of economic research to find better instruments to measure the potential output and the busi- ness cycle, and thus provide better guidance for the fiscal policy. 1.1 Aims of this project This project develops the methodology for measuring the potential output and assessing the medium-term growth prospects of the Finnish economy by forecasting the potential. We apply novel approaches to the estimation of the Finnish production function (constant elasticity of substitution, CES) and the filtration of the potential output (Sequential Monte Carlo method, SMC) with the aim at improving the European Commission (EC) production function method. Furthermore, we use structural macroeconomic model to estimate the medium-term growth potential of the Finnish economy. We base our measurement of the potential output on the production function method that is currently used by most economic institutions. Our starting point is the method of the Euro- pean Commission because of its important economic policy role. The aim of the project is to 1 It measures the government budgetary position when the effect of cyclical fluctuations and one-off expenditure and income items has been eliminated. 4 With the potential output of an economy, one refers to its output in the hypothetical case in which its output is not distorted by cyclical factors or random fluctuations. The standard definition of the potential is the level of output that an economy can produce without triggering above-normal inflation. The definition is originally introduced by Okun who emphasized that potential output is a “supply concept, a measure of pro- ductive capacity.” (Coibion et al., forthcoming). Being a theoretical concept, there is no direct way to measure the potential output, and one may (roughly) distinguish between three approaches to estimating and pre- dicting it (cf. Coibion et al., forthcoming). In what one could call a (purely) statistical approach, one tries to discern the potential output from past output data without introducing any particular theory on the factors which determine it or make the actual and potential output differ. In practice, the potential output is typically identified by estimating the low frequency movement of the actual production. The inflation-neutral growth of the economy is assessed by fore- casting the low-frequency trend of the economy and comparing it with actual econom- ic development. The methods used for measuring the trend are highly diverse (see e.g. Murray, 2014). In the simplest methods one uses univariate time series filters to isolate the variation of the time series that is considered to be due to the business cycle. This approach has merits in providing clear definitions of the business cycle, but it is problematic because the business cycles are not all similar and there is large variation in their frequencies and effects on economic activity. Alternatively, the trend growth can be estimated through multivariate methods based on, for example, the use of inflation (Phillips curve), unemployment (Okun’s law), the capacity utilization rate, and many other variables as auxiliary variables. The information used can be filtered by various filters (e.g. Hodrick-Prescott filter, bandpass filter) or other information-selecting meth- ods such as principal component analysis, or the advance aggregation of auxiliary variables into indices. The large variety of approaches has not yielded a unique solution to the problem of determining the potential output and the output gap. On the contrary, as Murray (2014) and several other studies show, different statistical methods produce varying results on the magnitude of the potential and the gap, and these results can deviate considerably from each other. Since the output gap is never observed in practice, the great unanswered question on trend measurements is related to model selection: on what grounds should the method, trend and eventually the output gap be selected? Secondly, potential output may be evaluated using the production function method. This method is used by many institutions (OECD, IMF, European Commission). Also the methodology of the European Commission (Havik et al., 2014), which makes use of a Cobb-Douglas production function, and the methods which are developed in Chapters 2 and 3 below and which are based on the CES function exemplify the production function approach. Box 1.1 How to measure potential output 4 5 In this approach one aggregates a comprehensive view of the production capacity of the economy (production function) which is based on economic theory and observa- tions of the state of various components. One tries to estimate the output as a func- tion of the inputs (including at least capital and labor) and the measure (or measures) of the productivity of the economy. The potential output may then at each moment of time be defined as the output which corresponds to the equilibrium level of the inputs and productivity at that time. If the actual dependence of output on various inputs and the equilibrium values of the inputs are well-known, this approach can help in esti- mating the output gap. On the other hand, the more uncertainty there is concerning the correct function and the equilibrium values, the less useful this approach is. In the worst-case scenario, false beliefs about the operations of economic mechanisms during, for instance, financial crises may lead to major biases in statistical inferences. To avoid the problem of selecting false theoretical foundations for the analysis, the empirical methods are typically constructed in a manner that offers flexibility in the choice of the model. For example, when separating the cyclical component of unem- ployment and the wage-inflation neutral potential (NAWRU), one may use empirical models which contain various known operating models of the labor market as special cases. However, the flexibility is not without problems. The more flexible a statistical model is, the more choices its user must make between economic theories. When evaluating the results, a statistician must decide whether the financial economic mechanism that are implied by the model are sensible, or whether the result is indica- tive of the problems associated with the statistical inference. Thirdly, it is possible to introduce, not just a production function, but a whole macro- economic model for the economy and deduce an estimate of the potential output from it. In this method, the potential output and potential growth projections are obtained by setting some random shocks and frictions to zero. This approach is exemplified by the use of ETLA model in Chapter 4. Besides the European Commission, the institutions which produce important esti- mates of potential output include the Congressional Budget Office (CBO) of the USA, the Federal Reserve Board, the International Monetary Fund (IMF) and the OECD. The assessments of the output gap by the Federal Reserve are judgemental (cf. Edge & Rudd, 2012, p. 2) in the sense that they are not based on any particular fixed model, while the methods used by the IMF are different for different countries (cf. de Resende, 2014, p. 24). Both the OECD and the CBO have developed production function methods of their own. It might be of some interest to contrast the methodology of the European Commis- sion (presented in Sections 2.2 and 2.7 below) with the methods of the CBO (cf. Shackleton, 2018). The CBO method divides the considered economy (i.e., the US as a whole) into six sectors (Nonfarm business, Farm, Household, Nonprofit, Feder- al government, and State and local government). The potential output is calculated separately and with partly different methods for the different sectors. Just like the EC method (see Chapter 3 below), the CBO method estimates potential labor supply using estimates of the participation rate, the equilibrium unemployment 6 rate, and the number of working hours per employee. The potential participation rate is estimated separately for 516 groups of people, which differ with respect to age, sex, race or ethnicity, and level of education. These determine the aggregate participation rate (and the number of persons in the labor market) of the economy, when they are combined with population data. Combined with the natural unemployment rate, they also determine the equilibrium labor supply when it is measured in persons rather than in hours. The potential weekly number of hours worked is estimated for each of the six sectors separately, using a relatively simple regression model. In addition, estimates of the potential employment share in each sector (i.e. the equilibrium share that the employ- ees of a sector have among all employed persons) are used in order to arrive at the potential labor supply of the economy. The production function is different in different sectors. The most important of the six sectors is the nonfarm business sector, which produces about 75% of the US GDP (Shackelton, 2018, p. 7). Its production function is a familiar Cobb-Douglas function, in which the potential output is determined by the potential labor supply, potential total factor productivity (TFP), and the capital stock. Also the potential TFP is estimated by a relatively simple regression model. Summing up, the CBO method is (unlike the EC method) based on dividing the labor force into groups and the economy into sectors, for which estimates of potential output are deduced separately. This makes the amount of data that the CBO method uses much larger, but the CBO method is nevertheless much simpler from the point of view of the mathematician, since it does not make use of any mathematical tools which would be as sophisticated as e.g. the Kalman filter which we describe in Sec- tion 2.7. All in all, due to both genuine uncertainty and the difficulty of choosing a proper estimation model, the potential output estimates often change considerably over time (see, for example, Orphanides & Van Norden, 2002; Rünstler, 2002; Planas & Ros- si, 2004; Golinelli, 2008; Marcellino & Musso, 2010; Bouis et al., 2012; Kuusi, 2015; Busse, 2016). The application of the Commission method during the Great Recession confirms the rule: The revisions have been large (Busse, 2016; Kuusi, 2018). Essen- tial forms of uncertainty include the uncertainty concerning the value of knowledge which has been accumulated in companies and individuals, and uncertainty concern- ing their broader operating environment, including institutions and infrastructure. In small open economies such as Finland, the problems of evaluation are particularly important, as the production is heavily influenced by links to foreign economies and the international competitiveness of production. The small open economies may also become more easily subjected to structural shocks as a result of their specialized production structures. 6 7 improve the production function method in order to provide better predictions of the potential and output gap in real time, and also more generally make the production function method more suitable for the Finnish economy. The method used by the European Commission raises several problems, and this project addresses a few remedies that aim to improve the functioning of the output gap methodology. We also use a structural macroeconomic model to project the potential output of the Finnish Economy over the medium term. The considered model is ETLA’s multi-sector growth model that incorporates forecasts of total-factor productivity changes, resources of the economy, as well as the conditions for external trade, and it produces a coherent forecast of the eco- nomic growth. 1.2 The main contributions Let us next introduce the main contributions of our analysis. The research outcomes I and II are based on the analysis conducted by the Labor Institute for Economic Research in section 2. The outcomes III and IV are based on the analysis of the Research Institute of the Finnish Economy in sections 3 and 4. I. We use the constant elasticity of substitution (CES) functions as the production function. This choice is more general than the Cobb-Douglas specification that is used by the European Commission and it allows us to distinguish between changes in labor aug- menting productivity and capital augmenting productivity which affect the total factor pro- ductivity (TFP). We develop a method for estimating the CES function from data on output, net capital stock, labor hours, wages and the user cost of capital. The estimation is carried out with different choices of the estimation model, and using also other capital and labor input series than the series which stem the plain national accounts data. Our results sug- gest that the CES production function is more appropriate for the Finnish economy than the Cobb-Douglas production function specification and that the elasticity between capital and labor is considerably smaller than what the Cobb-Douglas function would imply. II. We find that the distinction between labor augmenting productivity and capital augmenting productivity matters for the measurement of the production potential. The results indicate that capital augmenting productivity tends to develop in a more procycli- cal manner than the labor augmenting productivity. We show this by applying the European Commission (EC) TFP method individually to the both productivity series. It also turns out that procyclicality of the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. III. We apply novel filtration methods in order to further analyze the cyclical sensitivity of the elements of the production function. In particular, we apply the Sequential Monte Carlo (SMC) method to analyze the cyclical co-variance of the labor inputs and the fac- tor-augmenting productivities. In addition to the cross-variation of the different components of the production function, we also use various indicators of the business cycle as well as detailed Phillips curve specifications. We first apply the SMC method to investigate the potential level of the labor input. While we use the Phillips curve to identify the non-accelerating inflation rate of unemployment, we simul- taneously model the cross-variation of the cyclical components of unemployment and the labor- 8 force participation rate, as well as other cyclical factors to jointly identify the business cycle. Furthermore, we augment the EC Phillips curve with true, forward-looking expectations, and the anchored inflation expectations proposed by Rusticelli et al. (2015). In the estimation we use Bayesian estimation with the Commission estimates as the starting point for choosing priors. We find that the model identifies reasonable estimates for both the NAWRU and the poten- tial level of the labor force participation rate. While the estimates are similar to EC estimates during the Great Recession, we note that our estimates are considerably smoother during the Finnish Great Depression of the 1990s. Thus, our findings provide evidence against large and short-lived variation of the potential labor input during economic crises, which both the EC and the OECD estimates indicate. We find that the results of the model are not very sensitive to the choice of the Phillips curve, while they indicate that the EC methodology may be preferable to the OECD method. Finally, our results indicate that some restrictions to the signal-to-noise ratios of the filters are warranted to ensure that the potential estimates have theoretically desirable properties. As a second application of the SMC, we analyze the cyclical cross-variation of the capital and labor augmenting productivities. We find empirical evidence for the cross-variation due to the influence of the business cycle: when the capital-augmenting productivity falls due to the business cycle, the labor augmenting efficiency reacts by increasing. The most natural explanation is that when the capacity utilization of capital falls, it is accompanied by layoffs of the workers that tend to increase the average productivity of the continuing workers. All in all, the results of our SMC analysis suggest that the influence of the business cycle on the Finnish economic activity may have been larger than the European Commission method would suggests. We also find that especially the NAWRU and the labor augmenting effi- ciency are very sensitive to the real-time uncertainty, and therefore economic policy that is steered based on them should be cautious. IV. We also consider a structural macroeconomic model of the medium-term growth potential of the Finnish Economy in the years 2019–2023. The considered model is ETLA’s multi-sector growth model that incorporates forecasts of total-factor productivity changes, re- sources of the economy, as well as the conditions for external trade, and it produces a coher- ent forecast of the economic growth. We show that the baseline growth path of the model and its structural changes are rather well matched with the data in the years 2000 to 2017, which lends credibility to its projections concerning GDP volume growth also in the medium-term. According to the considered model, the forecasted average growth rate of the GDP volume growth is at 1.5% per annum in the years 2019–2023, when the model is employed with the long-term historical average total-factor productivity growth in the different sectors, the latest population forecasts, as well as the external market that matches with the actual shares of foreign imports in the domestic markets and the structure of the Finnish exports. By using counterfactual scenarios, we find that the economic growth is predominately determined by the technological improvements in the information and communications technology and the external trade. Finally, as the prior analysis of the labor and productivity potential suggests that they are frequently and heavily revised, we discuss growth scenarios with alternative productivity and employment paths. We find that there is a large variation in the possible potential growth rates of the Finnish Economy over the medium term. 8 9 2 CES AGGREGATE PRODUCTION FUNCTION AND THE OUTPUT GAP Sami Jysmä and Ilkka Kiema 2.1 Introduction The production function methodology which the European Commission uses for calculating output gaps is based on the Cobb-Douglas production function. The main elements of the method are (1) the filtration of the observed total factor productivity (TFP) to its trend and cy- clical components and (2) the analogous filtration of the observed labor supply. This section considers the Constant Elasticity of Substitution (CES) production function as an alternative to the Cobb-Douglas function, and discusses the former filtration (1) of the Commission in the context of the both functions. Chapter 3 discusses alternative filtration methodologies. Intuitively, the elasticity of substitution governs the substitution between capital and labor when the relative price of the inputs changes. The CES production function is more general than the Cobb-Douglas specification that is used by the European Commission. One may think of the Cobb Douglas function as the special case of the CES production function in which elasticity of substitution is exactly such that the nominal cost shares of the different factors remain constant. The generalization matters for several reasons. First, it turns out that the Cobb-Douglas function is not very suitable for Finland, and hence, by using the CES function, we avoid the possible errors that are due to the misspecification of the production function. Second, the CES class allows a distinction between labor augmenting productivity and capital augment- ing productivity. This contrasts with the single total factor productivity (TFP) series of the Cobb Douglas production function. As TFP is a key factor behind the movements of the potential output as well as the output gap, while its movements are still not very well under- stood, the distinction allows us to unravel the TFP black box. We develop a method for estimating the CES function from data on output, net capital stock, labor hours, wages and the user cost of capital. The estimation is carried out with different choices of the estimation model, and using also other capital and labor input series than the series which stem from the plain national accounts data. Our results suggest that the Cobb-Douglas production function specification can be rejected and that the elasticity be- tween capital and labor is considerably smaller than what the Cobb-Douglas function would imply. The results indicate that capital augmenting productivity tends to develop in a more procyclical manner than the labor augmenting productivity. We show this by applying the Eu- ropean Commission TFP method individually to both productivity series. It also turns out that procyclicality of the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 10 2.2 The Cobb-Douglas and the CES production function Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies EL and EK , but their actual contributions to the output is affected also by their degrees of utilization, UL and UK . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2) so that the total factor productivity may be written as (3) where (4) expresses the trend component and (5) expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, EL and EK , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utiliza- tion, UL and UK which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al., 1961, p. 230) (6) Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K / L with respect to 11 the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 2.2 The Cobb‐Douglas and the CES production function  Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) 1Y AL K  where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies LE and KE , but their actual contributions to the output is affected also by their degrees of utilization, LU and KU . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2)       1 1 1 1 L L K K L K L KY U LE U KE E E U U L K            so that the total factor productivity may be written as (3) A PC where (4) 1 L KP E E  expresses the trend component and (5) 1 L KC U U  expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, LE and KE , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utilization, LU and KU which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. 11 the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 2.2 The Cobb‐Douglas and the CES production function  Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) 1Y AL K  where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies LE and KE , but their actual contributions to the output is affected also by their degrees of utilization, LU and KU . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2)       1 1 1 1 L L K K L K L KY U LE U KE E E U U L K            so that the total factor productivity may be written as (3) A PC where (4) 1 L KP E E  expresses the trend component and (5) 1 L KC U U  expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, LE and KE , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utilization, LU and KU which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. 11 the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 2.2 The Cobb‐Douglas and the CES production function  Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) 1Y AL K  where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies LE and KE , but their actual contributions to the output is affected also by their degrees of utilization, LU and KU . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2)       1 1 1 1 L L K K L K L KY U LE U KE E E U U L K            so that the total factor productivity may be written as (3) A PC where (4) 1 L KP E E  expresses the trend component and (5) 1 L KC U U  expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, LE and KE , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utilization, LU and KU which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. 11 the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 2.2 The Cobb‐Douglas and the CES production function  Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) 1Y AL K  where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies LE and KE , but their actual contributions to the output is affected also by their degrees of utilization, LU and KU . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2)       1 1 1 1 L L K K L K L KY U LE U KE E E U U L K            so that the total factor productivity may be written as (3) A PC where (4) 1 L KP E E  expresses the trend component and (5) 1 L KC U U  expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, LE and KE , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utilization, LU and KU which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. 11 the potential output is reduced when the Cobb-Douglas production function is replaced by a CES function in the Commission’s methodology. 2.2 The Cobb‐Douglas and the CES production function  Formally, the production function used by the European commission is the familiar Cobb-Douglas function which may be written as (1) 1Y AL K  where Y is GDP, K is capital, L is labor, and A is total factor productivity. It is assumed that the factors of production L and K have the (trend) efficiencies LE and KE , but their actual contributions to the output is affected also by their degrees of utilization, LU and KU . More precisely, the commission’s methodology assumes that the output Y is given by (Havik et al., 2014, p. 10) (2)       1 1 1 1 L L K K L K L KY U LE U KE E E U U L K            so that the total factor productivity may be written as (3) A PC where (4) 1 L KP E E  expresses the trend component and (5) 1 L KC U U  expresses the cyclical component of the total factor productivity. The Cobb-Douglas production function does not allow for distinguishing between the effects that changes in the two efficiency factors, LE and KE , have on total factor productivity. As we shall see in Section 2.5. below, the production function methodology of the European Commission uses capacity utilization data for estimating the cyclical component C, but the method does not allow one to distinguish between variations in the two degrees of utilization, LU and KU which determine C. However, both distinctions can be drawn meaningfully in the context of the constant elasticity of substitution (CES) production function. 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 10 11 stays constant. More specifically, this elasticity has the value The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, the elasticity of substitution s approaches s=1 (cf. Arrow et al., 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was sug- gested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formu- lation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers Xt and Bt as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) Here Y is production, K is the stock of capital, L the flow of labor hours, and r and p0 are parameters. The factor substitution parameter r is related to the elasticity of substitution via (8) It is obvious that if the combination X = X0 , K = K0 , B = B0 , L = L0 , corresponded to the actual values of X, K, B, L at some point of time t = 0, the value of Y which the production function (7) yields for t = 0 would be Y = Y0. Hence, we may think of the set {Y0, X0, K0, B0, L0} as the normalization point of the function F. Also the parameter p0 has now a clear inter- pretation: formula (7) implies that if both capital and labor earned their marginal products at the normalization point, the capital income share would at that point be (9) and the labor income share would be (For a proof, see Appendix.) The estimates of s have been seen to convergence across studies. In particular, accord- ing to two recent surveys on the subject, the evidence points to a substitution elasticity of 0.4–0.6 for the U.S. economy (Chirinko, 2008) and similar values for a number of other de- 2 To prove this directly, on should express the CES production function (6) in the form , take logarithms and use l’Hôpital’s rule in the limit in which . 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 12 The CES production function was first introduced into the literature by Solow (1956, p. 77), who famously drafted the function in order to “provide one bit of variety” into his theoretical investigation of economic growth. Five years later the production function was named and further refined by Arrow et al. (1961), who also were the first to apply the function in an empirical setting. In its original form, the CES function could be written simply as (cf. Arrow et al, 1961, p. 230) (6)  1/ pp pY aK bL  Its defining characteristic, constant elasticity of substitution, means (see Klump et al., 2012, p. 773) that the elasticity of capital intensity K Lwith respect to Y L Y K     stays constant. More specifically, this elasticity has the value 1 1 p    The Cobb-Douglas production function can be viewed as the limiting case of (6) in which p approaches 0 so that, in accordance with (8), the elasticity of substitution  approaches 1  (cf. Arrow, 1961, p. 231).2 Unlike the Cobb-Douglas function, the function (6) allows one to separate capital augmenting and labor augmenting changes of the production function, as these may be represented as changes in a and b, respectively. However, the parameters a and b of the function (6) lack any obvious economic interpretation, and their values will depend on the choice of units for capital and labor. The normalization procedure which was suggested in Chirinko et al. (2011) and Klump et al. (2007) provides a more intuitive formulation for the CES function, and it leads to an estimation procedure for it. Introducing the multipliers tX and tB as representations of the capital augmenting and labor augmenting technological change, the normalized production function may be written as (cf. Klump et al., 2012, p. 779) (7) 1 0 0 0 0 0 0 0 ( , , , ) (1 )X K B LY F X K B L Y X K B L                          2 To prove this directly, on should express the CES production function (6) in the form   11 pp pY K L     , take logarithms and use l’Hôpital’s rule in the limit in which 0p . 13 Here Y is production, , K is the stock of capital, L the flow of labor hours, and  and 0 are parameters. The factor substitution parameter  is related to the elasticity of substitution via (8) 1     It is obvious that if the combination 0 0 0 0, , ,X X K K B B L L    corresponded to the actual values of , , ,X K B L at some point of time 0t  , the value of Y which the production function (7) yields for 0t  would be 0Y Y Hence, we may think of the set  0 0 0 0 0, , , ,Y X K B L as the normalization point of the function F. Also the parameter 0 has now a clear interpretation: formula (7) implies that if both capital and labor earned their marginal products at the normalization point, the capital income share would at that point be (9)   00 0 0 Y K K Y     and the labor income share would be   0 0 0 1 Y L L Y      (For a proof, see appendix 1.) The estimates of  have been seen to convergence across studies. In particular, according to two recent surveys on the subject, the evidence points to a substitution elasticity of 0.4 – 0.6 for the U.S. economy (Chirinko, 2008) and similar values for a number of other developed countries (Klump, Mcadam, & Willman, 2012). In the case of Finland, to the best of our knowledge, there exists three publications that have estimated the elasticity with Finnish data: Ripatti & Vilmunen (2001), Jalava et al. (2006) and Luoma & Luoto (2010), with substitution elasticity estimates of around 0.6, 0.4 – 0.5 and 0.5 respectively. 2.3 An estimation procedure for the normalized CES function  13 Here Y is production, , K is the stock of capital, L the flow of labor hours, and  and 0 are parameters. The factor substitution parameter  is related to the elasticity of substitution via (8) 1     It is obvious that if the combination 0 0 0 0, , ,X X K K B B L L    corresponded to the actual values of , , ,X K B L at some point of time 0t  , the value of Y which the production function (7) yields for 0t  would be 0Y Y Hence, we may think of the set  0 0 0 0 0, , , ,Y X K B L as the normalization point of the function F. Also the parameter 0 has now a clear interpretation: formula (7) implies that if both capital and labor earned their marginal products at the normalization point, the capital income share would at that point be (9)   00 0 0 Y K K Y     and the labor income share would be   0 0 0 1 Y L L Y      (For a proof, see appendix 1.) The estimates of  have been seen to convergence across studies. In particular, according to two recent surveys on the subject, the evidence points to a substitution elasticity of 0.4 – 0.6 for the U.S. economy (Chirinko, 2008) and similar values for a number of other developed countries (Klump, Mcadam, & Willman, 2012). In the case of Finland, to the best of our knowledge, there exists three publications that have estimated the elasticity with Finnish data: Ripatti & Vilmunen (2001), Jalava et al. (2006) and Luoma & Luoto (2010), with substitution elasticity estimates of around 0.6, 0.4 – 0.5 and 0.5 respectively. 2.3 An estimation procedure for the normalized CES function  13 Here Y is production, , K is the stock of capital, L the flow of labor hours, and  and 0 are parameters. The factor substitution parameter  is related to the elasticity of substitution via (8) 1     It is obvious that if the combination 0 0 0 0, , ,X X K K B B L L    corresponded to the actual values of , , ,X K B L at some point of time 0t  , the value of Y which the production function (7) yields for 0t  would be 0Y Y Hence, we may think of the set  0 0 0 0 0, , , ,Y X K B L as the normalization point of the function F. Also the parameter 0 has now a clear interpretation: formula (7) implies that if both capital and labor earned their marginal products at the normalization point, the capital income share would at that point be (9)   00 0 0 Y K K Y     and the labor income share would be   0 0 0 1 Y L L Y      (For a proof, see appendix 1.) The estimates of  have been seen to convergence across studies. In particular, according to two recent surveys on the subject, the evidence points to a substitution elasticity of 0.4 – 0.6 for the U.S. economy (Chirinko, 2008) and similar values for a number of other developed countries (Klump, Mcadam, & Willman, 2012). In the case of Finland, to the best of our knowledge, there exists three publications that have estimated the elasticity with Finnish data: Ripatti & Vilmunen (2001), Jalava et al. (2006) and Luoma & Luoto (2010), with substitution elasticity estimates of around 0.6, 0.4 – 0.5 and 0.5 respectively. 2.3 An estimation procedure for the normalized CES function  12 veloped countries (Klump et al., 2012). In the case of Finland, to the best of our knowledge, there exists three publications that have estimated the elasticity with Finnish data: Ripatti & Vilmunen (2001), Jalava et al. (2006) and Luoma & Luoto (2010), with substitution elasticity estimates of around 0.6, 0.4–0.5 and 0.5 respectively. 2.3 An estimation procedure for the normalized CES function As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values Kt , Lt , and Yt are available at some points of time t = 1,...,T. Following the lead of Klump et al. (2012), we shall not choose t = 0 to correspond to some actual point of time for which observations are available. Rather, we define and where denotes the sample geometric mean of each series . It immediately follows from these definitions and (7) that However, this formula does not provide us with a method of estimating Y0 , since due to the nonlinearity of the CES function, is not necessarily identical with (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier z which connects and via (10) Analogously with our earlier characterization (9) of p0 we now define when t = 1,...,T and when is calculated for the values X = Xt , K = Kt , B = Bt and L = Lt , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be pt and 1 – pt if they both earned their marginal prod- ucts. It can also be shown that (11) and that (12) where is the geometric mean of the capital share pt when t = 1,...,T and similarly, is the geometric mean of the labor share 1 – pt (see Appendix for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as (13) 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     and that (12) 1      where  is the geometric mean of the capital share t when 1,...,t T and similarly, 1  is the geometric mean of the labor share 1 t (see Appendix I for derivations of the above results). Combining (7), (10), and (11), we now write the estimable version of our production function as 14 As our next step, we shall specify an estimation procedure for a CES function of the form (7). We assume that the observed values tK , tL , and tY are available at some points of time 1,...,t T . Following the lead of Klump (2012), we shall not choose 0t  to correspond to some actual point of time for which observations are available. Rather, we define 0 0 0, ,X X K K B B   and 0L L where Z denotes the sample geometric mean of each series tZ . It immediately follows from these definitions and (7) that 0( , , , )F X K B L Y However, this formula does not provide us with a method of estimating 0Y , since due to the nonlinearity of the CES function, ( , , , )F X K B L is not necessarily identical with Y (the geometric mean of the observed outputs). Accordingly, we now introduce a multiplier  which connects 0Y and Y via (10) 0Y Y Analogously with our earlier characterization (9) of 0 we now define   tt t t Y K K Y     when 1,...,t T and when  tY K  is calculated for the values , ,t t tX X K K B B   and tL L , which correspond to the point of time t. This definition means that the income shares of capital and labor at time t would be t and 1 t if they both earned their marginal products. It can also be shown that (11) 0     a