Substitution Elasticities in a CES Production Framework · 2012. 5. 9. · Kmenta vs. Non-linear Estimations CES functions can also estimated with linear estimation techniques e.g

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


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


� 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




� 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)





� 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 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

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


9

� 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


1. Estimation

2. Estimation

10

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


11

� 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)


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


not Cobb-Douglas

not Leontief

at p<0,01

13

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


( )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


of the sectors

• rejected for 1of

the sectors

• rejected for all

but two sectors


but three sectors


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


( )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


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


16

� 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]

17

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

19

Top and Middle

Nest

Bottom Nest

Appendix - Sectors


20

Appendix – Results 1

Leontief or Cobb-Douglas


21


Change over time


22


Substitution Elasticities (PORT, restricted, with starting values)


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Documents

Substitution Elasticities in a CES Production Framework · 2012. 5. 9. · Kmenta vs. Non-linear Estimations CES functions can also estimated with linear estimation techniques e.g