Substitution Elasticities in a CES Production Framework An Empirical Analysis on the Basis of Non-Linear Least Squares Estimations Simon Koesler and Michael Schymura Zentrum für Europäische Wirtschaftsforschung (ZEW) Final WIOD Conference – 24.4.2012, Groningen 1
Outline 1. Motivation and Objective 2. State of Research 3. Data and Estimation Procedure 4. Results 4. Results 2
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Motivation and Objective � A multitude of challenges call for regulative interventions by policy makers, e.g. global warming or trade issues � In particular in times of turbulent economic outlook and scarce resources, effectiveness, cost-efficiency and distribution issues are crucial � Need for capable and reliable instruments to assess regulations ex ante � CGE models have become an important instrument in evaluating alternative policy measures � Elasticities are key parameters for CGE models as they are critical for determining the comparative static behaviour of the models � CGE models build frequently on CES functions � Our objective is to provide modellers with the required elasticities 3
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results CES Framework � Constant Elasticities of Substitution functions (CES) have become the backbone of CGE models; eg. single-nest two input CES 1 − σ ( ) 1 ( ) ) ( ) − ρ ( − ρ λ t = γ α + − α γ ≥ ≤ α ≤ ρ = ≥ − 1 − ρ 0 ; 0 1 ; 1 Y e X X t 1 ; t 2 ; t σ � We investigate input substitutability in a three level KLEM CES production Top Nest Top Nest framework of the form: Substitution Elasticity Middle Nest � How does the ratio of inputs change if the ratio of their marginal product changes? Bottom Nest 4
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results State of Research 1/4 Kemfert (1998) � Studies substitution elasticities between K, L, E for three production structures (KLE, KEL, LEK) � � Estimation: directly from CES function; non-linear least squares Estimation: directly from CES function; non-linear least squares � Data: time series data for German industry (German statistical office) � CES framework is adequate to characterise German industry � KLE production structure provides best fit � provides estimates for 7 German sectors 5
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results State of Research 2/4 van der Werf (2008) � Investigates input substitutability between K, L, E for three production structures (KLE, KEL, LEK) � � Estimation: cost-function approach, linear least squares Estimation: cost-function approach, linear least squares � Data: IEA Energy Balances, OECD International Sectoral Database � confirms usage of KLE production structure � general use of Cobb-Douglas functions is too simplistic � provide estimates for 7 sectors 6
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results State of Research 3/4 Okagava and Ban (2008) � Study input substitutability between K, L, E and M, S for a production structure of the form KLE(MS) � � Estimation: cost-function approach, linear least squares Estimation: cost-function approach, linear least squares � Data: EU-KLEMS � provide estimates for 22 sectors � has become very popular among modellers because of its comprehensive sectoral coverage and consideration of intermediates and services 7
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results State of Research 4/4 � Substitution parameters for CGE models are estimated using directly a CES production function in the estimation process � But so far, in particular for substitution elasticities in the CES framework only few estimates of the required elasticities exist and those available … � are limited to a narrow set of sectors � rely on a combination of originally unrelated data sources � focus on substitutability between specific inputs � build on linear-estimations (Kmenta or cost function approach) � We estimate elasticities of substitution directly in the framework of a three-level nested KLEM CES production function for 34 sectors 8
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Estimation Procedure 1/2 Data � World-Input-Output Database (WIOD) � WIOD: covers 34 sectors; includes 40 regions (EU, BRIC, USA, etc.); offers annual data for the period 1995 - 2009 � � Variables used Variables used � Benefits of WIOD: + data can be derived from one consistent dataset + comprehensive sectoral coverage + dataset can also be used to calibrate models 9
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Estimation Procedure 2/2 Estimation � Estimate substitution elasticities directly on the basis of an estimation equation having the form of a CES production function � Non-linear least squares estimation using different optimisation algorithms, in parts restricted (CES side-constraints) and with starting values from a preceding grids-search � Substitution elasticities are estimated individually for each sector; initially 2. Estimation polled across time and regions � No need for price data 1. Estimation � No need for Kmenta approximation 10
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Results 1/5 Basic Findings � Overall estimations are robust across all estimation techniques � We concentrate on optimisation algorithms with the best fit and convergence, i.e. PORT routines � � Expectedly starting values from a grid search increase the fit and ease Expectedly starting values from a grid search increase the fit and ease convergence � For a small set of elasticities, unrestricted estimations provide results − σ 1 violating the basic CES framework γ ≥ ≤ α ≤ ρ = ≥ − 0 ; 0 1 ; 1 σ � Estimation procedures allowing to restrict parameters should be used in this context � Indication that for a small set of sectors CES framework might not be ideal 11
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Results 2/5 Kmenta vs. Non-linear Estimations � CES functions can also estimated with linear estimation techniques e.g. by using Kmenta approximations transforming CES functions in a linear system � In the KL nest, for all sectors non-linear estimations perform better (fit to data) � Estimations using Kmenta approximations tend to underestimate ρ KL compared to non-linear estimations using PORT routines Kmenta PORT ρ KL ρ KL R 2 R 2 Std.Dev. Std.Dev. TRN Equ. 0,79 1,40 0,18 4,51 1,32 0,97 Air TRN -0.27 0,24 0,92 0,76 0,40 0,95 12
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Results 3/5 Leontief or Cobb-Douglas not Leontief not Cobb-Douglas at p<0,01 13
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Results 4/5 Changes in Input Substitutability over Time � In principle, technological progress could also take place by means of changes in input substitutability over time � In an extended CES framework this implies that ρ is time dependent: 1 − − ρ ∑ ∑ ( ( ) ) ρ t = = γ γ α α Y Y X X t t t t t i i it it i • rejected for 1/3 • rejected for 1/3 • rejected for 1of of the sectors of the sectors the sectors • rejected for all • rejected for all • rejected for all but two sectors but three sectors but two sectors � Input substitutability appears to change over time � But between 1997 and 2007 changes appear to be rather small for the majority of sectors 14
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results Results 5/5 Changes in Input Substitutability across Regions (Work in Progress) � Originally, we did not control for potential regional differences � But ρ could vary across regions 1 − ρ ∑ ∑ ( ( ) ) − ρ r = = γ γ α α Y Y X X r t t t t i i it it i • rejected for • rejected for 8/33 • rejected for 17/23 of sectors of sectors 21/35 of sectors • rejected for • rejected for • rejected for 21/23 of sectors 24/33 of sectors 33/35 of sectors � Input substitutability changes only for few sectors across regions, although σ KLE is an exception � Changes across regions tend to be rather small 15
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