Substitution Elasticities in a
CES Production Framework
An Empirical Analysis on the Basis of
Non-Linear Least Squares Estimations
1
Simon Koesler and Michael Schymura
Zentrum für Europäische Wirtschaftsforschung (ZEW)
Final WIOD Conference – 24.4.2012, Groningen
1. Motivation and Objective
2. State of Research
3. Data and Estimation Procedure
4. Results
Outline
4. Results
2
� 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
Motivation and Objective
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
� 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
CES Framework
Top Nest
� Constant Elasticities of Substitution functions (CES) have become the
backbone of CGE models; eg. single-nest two input CES
� We investigate input substitutability in a
three level KLEM CES production
framework of the form:
( ) ( )( )( ) 11
;10;01
1
;2;1−≥
−=≤≤≥−+= −−−
σ
σραγααγ ρρρλ
tt
t
tXXeY
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
4
Bottom Nest
Middle Nest
Top Nestframework of the form:
Substitution Elasticity
�How does the ratio of inputs change
if the ratio of their marginal product
changes?
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
State of Research 1/4
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
� 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
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
State of Research 2/4
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
� 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
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
State of Research 3/4
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
� 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
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� 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
State of Research 4/4
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
� 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
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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
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
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� Variables used
� Benefits of WIOD: + data can be derived from one consistent dataset
+ comprehensive sectoral coverage
+ dataset can also be used to calibrate models
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 in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
1. Estimation
2. Estimation
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values from a preceding grids-search
� Substitution elasticities are estimated
individually for each sector; initially
polled across time and regions
� No need for price data
� No need for Kmenta approximation
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
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
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� Expectedly starting values from a grid search increase the fit and ease
convergence
� For a small set of elasticities, unrestricted estimations provide results
violating the basic CES framework
� 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
−≥
−=≤≤≥ 1
1;10;0
σ
σραγ
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)
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
12
data)
� Estimations using Kmenta approximations tend to underestimate ρKL
compared to non-linear estimations using PORT routines
Kmenta PORT
ρKL Std.Dev. R2 ρKL Std.Dev. R2
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
Results 3/5
Leontief or Cobb-Douglas
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
not Cobb-Douglas
not Leontief
at p<0,01
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at p<0,01
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:
Results 4/5
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
( )t
t
itittXY
ρρ
αγ−
−
= ∑1
14
� Input substitutability appears to change over time
� But between 1997 and 2007 changes appear to be rather small for the
majority of sectors
( )i
itittXY αγ
= ∑
• rejected for 1/3
of the sectors
• rejected for 1/3
of the sectors
• rejected for 1of
the sectors
• rejected for all
but two sectors
• rejected for all
but three sectors
• rejected for all
but two sectors
Changes in Input Substitutability across Regions (Work in Progress)
� Originally, we did not control for potential regional differences
� But ρ could vary across regions
Results 5/5
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
( )r
r
itittXY
ρρ
αγ−
−
= ∑1
15
� Input substitutability changes only for few sectors across regions,
although σKLE is an exception
� Changes across regions tend to be rather small
( )i
itittXY αγ
= ∑
• rejected for
17/23 of sectors
• rejected for 8/33
of sectors
• rejected for
21/35 of sectors
• rejected for
21/23 of sectors
• rejected for
24/33 of sectors
• rejected for
33/35 of sectors
Conclusion
Summary
� Non-linear estimation techniques outperform estimations using Kmenta
approximations
� Neither Cobb-Douglas nor Leontief functions are adequate
approximations of sectoral production behaviour
Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results
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� Although changes over a period of 12 years are rather minor,
substitution elasticities may change over time
Open Research Questions
� Do substitution elasticities vary across regions?
Simon Koesler
Telephone: +49 621 1235 203
Mail: [email protected]
Michael Schymura
Telephone: +49 621 1235 202
Mail: [email protected]
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References
� KEMFERT, C. (1998): Estimated substitution elasticities of a nested CES production function approach
for Germany, Energy Economics, Vol. 20, pp. 249-264
� VAN DER WERF, E. (2008): Production functions for climate policy modeling: An empirical analysis,
Energy Economics, Vol. 30, pp. 2964-2979
� OKAGAWA, A. and BAN, K. (2008): Estimation of substitution elasticities for CGE models, Discussion
Papers in Economics and Business, No. 08-16
Substitution Elasticities for CGE Models – Appendix
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Appendix - Estimation Framework
Substitution Elasticities for CGE Models – Motivation and Objective / State of Research / Data / Estimation Procedure / Results
Top and Middle
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Top and Middle
Nest
Bottom Nest
Appendix - Sectors
Substitution Elasticities for CGE Models – Appendix
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Appendix – Results 1
Leontief or Cobb-Douglas
Substitution Elasticities for CGE Models – Appendix
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Appendix – Results 2
Change over time
Substitution Elasticities for CGE Models – Appendix
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Appendix – Results 3
Substitution Elasticities (PORT, restricted, with starting values)
Substitution Elasticities for CGE Models – Appendix
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