1. Estimation of substitution elasticities in three-factor production functions: Identifying the role of energy
Julius Frieling, Reinhard Madlener
International Exergy Economics Workshop
University of Sussex, 15 July 2016

2. Energy demand responds to change in all input factors.
Introduction
Energy demand responds to change in all input factors.
Identifying the elasticity of substitution between energy, capital, and labor is important for understanding policy impact.
Existing estimation methods for CES functions are limited, in particular when dealing with more than two factors.
We find a practical solution to estimate CES functions and explain how elasticities should be interpreted.
We describe the implications of the estimated low elasticity of energy.

3. Introduction
CES functions are a good way to analyze aggregated production data, but they are hard to estimate.
When the analysis deals with more than two factors, getting robust estimates becomes much more difficult.
We adapt a method of estimating a system of equations in a stepwise procedure to improve the results.
Our results show the complex relation between the input factors, in particular between capital and energy inputs.

4. The Model
We use a general, normalized three-factor CES function as our model:
𝑌=𝜓 𝑌 0 𝛾 𝑉 𝑉 𝜎−1 𝜈 𝜈−1 𝜎 + 𝛾 𝐸 𝑒 𝛼 𝐸 𝑡− 𝑡 𝑄 𝐸 𝐸 𝐸 𝜎−1 𝜎 𝜎 𝜎−1
𝑉= 𝛾 𝐿 𝑒 𝛼 𝐿 𝑡− 𝑡 𝐿 𝐿 𝜈−1 𝜈 + 1− 𝛾 𝐿 𝑒 𝛼 𝐾 𝑡− 𝑡 𝐾 𝐾 𝜈−1 𝜈
Normalized function helps with improving estimation
Real user cost and output are related through the accounting identity:
𝑌≡𝑤𝐿+𝑟𝐾+𝑝𝐸
For the estimation of the parameters construct a system of equations containing the log of the prod. function and the logs of FOCs (derivation not shown):
log 𝑌 𝑌 0 = log 𝜓 + log (…) log 𝑤 = log 𝜕𝑌 𝜕𝐿 log 𝑟 = log 𝜕𝑌 𝜕𝐾 log 𝑝 = log 𝜕𝑌 𝜕𝐸

5. Suggested nested CES formulation by Sato (1967):
Fundamentals
Suggested nested CES formulation by Sato (1967):
Elasticity is determined within the a given CES process
Introduces new problem: How do we define elasticity?
Classic definition is Hicks-McFadden, it only considers two factors (-> implicitly holds other factors constant)
Allen-Uzawa elasticity frequently used, allows other factors to adjust (Always symmetrical ).

6. Fundamentals
Morishima elasticity (proposed by Blackorby, Russell) offers an alternative:
Elasticity as a measure of reaction to a change in relative prices.
Measured intraprocess and interprocess elasticities are Morishima
Direct Morishima elasticities can depend on factor shares within processes.
Nesting structure makes explicit assumptions about the structure of the process and separability.
for goods i, j in processes m,n (after Anderson, Moroney 1993):
𝜎 𝑖𝑗 = 𝜎 𝑗𝑖 = 𝜎 𝑚 𝑛 if 𝑖,𝑗∈𝑚 𝑛 𝜎 𝑖𝑗 = 𝛾 𝑖 𝑚 𝜎 𝑁 + 1− 𝛾 𝑖 𝑚 𝜎 𝑚 if 𝑖∈𝑚 and 𝑗∈𝑛 𝜎 𝑗𝑖 = 𝛾 𝑗 𝑛 𝜎 𝑁 + 1 − 𝛾 𝑗 𝑛 𝜎 𝑛 if 𝑖∈𝑚 and 𝑗∈𝑛
Where 𝜎 𝑁 is the estimated elasticity between the different CES nests and 𝜎 𝑚 𝑛 is the elasticity of process m(n).
However, elasticities between factors from different CES processes within the function have to be calculated explicitly and are not constant and asymmetric!

7. Empirical Results
Data Overview

8. The strategy is to build up the estimation and validate each step.
Empirical Results
The strategy is to build up the estimation and validate each step.
Start with two production factors:
Directly estimate the two-factor CES function
System approach and total factor productivity
Compare with results to system approach and identical factor prod.
System approach and generalized productivity
Expand the production function to three factors
Total factor prod. and estimate 𝜈
Identical factor prod. and estimate 𝜈
Generalized factor prod. and estimate 𝜈
Cobb-Douglas process for labor and capital
Identical factor prod. and set 𝜈=0.5598
Generalized factor prod. and set 𝜈=0.5598

9. Empirical Results Elasticity E – K/L Elasticity K – L
General productivity
Labor productivity
Capital productivity
Energy productivity
Morishima elasticity L – E
Morishima elasticity K – E

10. Fixing 𝜈=0.5598 has a tremendous impact on the results.
Empirical Results
Fixing 𝜈= has a tremendous impact on the results.
Sato (1967) already suggested a stepwise estimation for nested CES production functions.
We evaluate the validity of the results by using different fixed values in the nested process.
Values chosen to replicate certain situations and to assess behavior around estimated elasticity in two factor case:
𝜈 ∈{0.2, 0.5, , 0.66, 0.75, 0.99}

11. Holding nested elasticity constant significantly improves behavior
Conclusions
System approach allows for identification even without assumptions on technical change, but precautions are necessary.
Holding nested elasticity constant significantly improves behavior
Nesting structure not constrained enough by the system to ensure convergence in numerical estimation.
This can not be avoided by evaluating relative movements in price.
Empirical results point to correct nesting structure of factors

12. Productivity gains in K and L are energy-using.
Conclusions
Productivity gains in K and L are energy-using.
Results underpin crucial role of energy in the production process.
Capital productivity is linked to energy use.
Low elasticity energy influences behavior in two factor case.
In our case this is by construction.
Accounting identity must still hold in two factor case.
How does the elasticity and productivity of energy react to developments in capital stock?

13. Julius Frieling jfrieling@eonerc.rwth-aachen.de
You can find a version of this paper at
Julius Frieling

14. 𝑌=𝐴 𝑖=1 𝑛 𝛾 𝑖 𝑥 𝑖 −𝜌 − 1 𝜌 with 𝑖=1 𝑛 𝛾 𝑖 =1
Fundamentals
The basic CES function:
𝑌=𝐴 𝛾 𝑥 1 −𝜌 + 1−𝛾 𝑥 2 −𝜌 − 1 𝜌
Factor shares 𝛾∈ 0,1 , elasticity 𝜎=1/(1+𝜌)
Powerful tool, but how can it be expanded?
𝑌=𝐴 𝑖=1 𝑛 𝛾 𝑖 𝑥 𝑖 −𝜌 − 1 𝜌 with 𝑖=1 𝑛 𝛾 𝑖 =1
Only identical elasticities, which is very limiting

15. Empirical Results

16. Fundamentals
Estimating CES is usually done with the Kmenta approximation
Kmenta approximation is a Taylor approximation around 𝜎=1
León-Ledesma et al. show the strong resulting estimation bias.
Hoff shows Kmenta approximation cannot be applied to expanded CES function without limiting parameters
The “impossibility theorem” claims estimating a generalized productivity parameter and the elasticity simultaneously is impossible.
System approach is a solution, but can it be applied to three factor production functions?

17. The Model
log 𝑌 𝑌 0 = log 𝜓 + \sigmalog (…) log 𝑤 = log 𝜕𝑌 𝜕𝐿 log 𝑟 = log 𝜕𝑌 𝜕𝐾 log 𝑝 = log 𝜕𝑌 𝜕𝐸