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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 fr Europische Wirtschaftsforschung (ZEW) Final WIOD Conference

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Substitution Elasticities in a CES Production Framework

An Empirical Analysis on the Basis of Non-Linear Least Squares Estimations

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Simon Koesler and Michael Schymura Zentrum für Europäische Wirtschaftsforschung (ZEW) Final WIOD Conference – 24.4.2012, Groningen

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  • 1. Motivation and Objective
  • 2. State of Research
  • 3. Data and Estimation Procedure
  • 4. Results

Outline

  • 4. Results

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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

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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

( )

( )(

)

( )

1 1 ; 1 ; 1

1 ; 2 ; 1

− ≥ − = ≤ ≤ ≥ − + =

− − −

σ σ ρ α γ α α γ

ρ ρ ρ λ t t t t

X X e Y

Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results

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Bottom Nest Middle Nest Top Nest

framework of the form: Substitution Elasticity How does the ratio of inputs change if the ratio of their marginal product changes?

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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

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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

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SLIDE 7

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

  • nly 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

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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

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

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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 ; 1 ; σ σ ρ α γ

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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

Substitution Elasticities in a CES Framework – Motivation and Objective / State of Research / Data and Estimation / Results

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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

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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

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at p<0,01

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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

it i t t

X Y

ρ ρ

α γ

− − 

   =

∑

1

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Input substitutability appears to change over time But between 1997 and 2007 changes appear to be rather small for the majority of sectors

( )

i it i t t

X Y α γ     =

∑

  • rejected for 1/3
  • f the sectors
  • rejected for 1/3
  • f 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

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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

it i t t

X Y

ρ ρ

α γ

− −

    =

∑

1

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Input substitutability changes only for few sectors across regions, although σKLE is an exception Changes across regions tend to be rather small

( )

i it i t t

X Y α γ     =

∑

  • rejected for

17/23 of sectors

  • rejected for 8/33
  • f 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

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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?
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Simon Koesler Telephone: +49 621 1235 203 Mail: koesler@zew.de Michael Schymura Telephone: +49 621 1235 202 Mail: schymura@zew.de

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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

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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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