<p>Using the following CES function I backed out factor-augmenting indices $A_1$ and $A_2$ using OECD country level macro data. This process is akin to Solow's (1956) method for deriving technology residuals. I want to examine the bias in the direction of technical change by examining the temporal evolution of the factor-augmenting indices.</p>

<p>$$Y=[((A_1L^\alpha) K^{(1-\alpha)})^\sigma+ A_2E^\sigma]^{1/\sigma}$$</p>

<p>The elasticity of substitution parameter $\sigma$ is 0.1 (Estimated using MLE). $Y$ is GDP. $E$ is energy flow.</p>

<p>There is no evidence of trend in the two technical change indices until the last few decades of the sample. From there on, the labour-augmenting index grows rapidly. The other index augmenting energy declines gradually. This divergence in the indices coincides with the sudden energy price shocks. </p>

<p>It might look pretty obvious that there was a labour saving and energy using bias in technical change. Though, I am stuck at this point, <strong><em>how would I link the derived factor-augmenting indices with factor prices and factor ratios to impart the technical change biases in a systematic way?</em></strong></p>

<hr>

<p>I tried a slightly different formulation because of the nesting of the Cobb-Douglas function in CES. The evidence pointed out by <code>@luchonacho</code> 
 in previous posts shows that Cobb-Douglas is not an apropriate fit to the data given that there is little potential for $K$ and $L$ to be perfect substitutes.  </p>

<p>An alternative formulation could be 
$$Y=[(A_lL^\nu+ A_kK^\nu)^\frac{\sigma}{\nu}+ A_eE^\sigma]^{1/\sigma}$$
However, estimating $\nu$ and $\sigma$ would not be easy due to the nonlinearities in the relationship. Thus, the first formulation was adopted. Any suggestions welcome! Alternative formulation, methods etc.</p>

CC BY-SA 3.0
2017-10-13T12:08:43.330Z
Interesting research there. Some comments/questions: (i) what is your estimate of $\sigma$? The interpretation of labour-saving/augmenting depends on its value. (ii) have you tested for changes in $\sigma$? (iii) ideally you want your production function to be [normalised](https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf?9fcd5fddf87498417164716f2bbee9fd). (iv) why assuming elasticity of substitution between capital and labour equal to one? Most of the evidence I have seen point toward $\sigma_{K,L}<1$. Maybe use a translog production function. (v) can you clarify your question?
luchonacho
CC BY-SA 3.0
2017-10-13T13:12:26.927Z
@luchonacho Many thanks, I've edited the post. Is it good enough?
london
CC BY-SA 3.0
2017-10-13T13:18:53.223Z
0.01? Really? So basically, no substitution between K-L bundle and energy? is this in line with other research? As far as I know the translog allows for quite a lot of freedom in terms of substitution. You do not face the nonlinearity problem.
luchonacho
CC BY-SA 3.0
2017-10-13T14:49:02.927Z
Sorry, I hastily typed the original edit and ended up making a number of typos due to having to step out of my office. I've edited the post again. The elasticiy value is the average for several countries. Please have a look.
london
CC BY-SA 3.0
2017-10-13T15:03:50.707Z
What is Y in your data? Notice value added might not be a good proxy for Y, because this measures contribution from non-contemporaneously produced factors of production and not from intermediate inputs like energy, which are also produced by K and L. This is, $E=f(K_E,L_E)$, both of which are included in $K$ and $L$.
luchonacho
CC BY-SA 3.0
2017-10-13T15:35:11.843Z
It is GDP. Yes, I am aware of this and this exercise is my inital attempt at working out a strategy for pinning down the bias in technical change. I think, this will be a limitation for this work.The production function is not normalised either. So, will work on it for a few months. I will try translog function now. Also, updated the post.
london
CC BY-SA 3.0
2017-10-13T16:29:22.087Z
I still do not understand the question. What else do you want to estimate? Why are your estimations of $A_i$ not sufficient for your interest?
luchonacho
CC BY-SA 3.0
2017-10-13T16:51:22.287Z
Please note that $A_i$s are time varying indices, I should have typed them as $A_{i,t}$, my bad. Surely, the indices show the turns and twists that indicate possible bias, but I want to pin down where the bias stems from. For now, I am looking at the correlation between $A_{i,t}$s and energy price and wages. There is a positive correlation between $A_{e,t}$ and energy prices indicating energy saving-bias induced by energy shocks. But not sure if this approach is sufficient to make a conclusion. Also, please see the question in the post highlighted in bold.
london
CC BY-SA 3.0
2017-10-14T09:15:49.913Z
So you do not want to pin down the bias but the source of the bias?
luchonacho
CC BY-SA 3.0
2017-10-14T10:25:31.933Z
Yes, as I note in the last comment I can decipher the bias from the trend breaks in technology residuals, but also want to identify the sources of the bias. Though, I do not have clear strategy for this - just looking at various approaches. Any suggestions welcome.
london
CC BY-SA 3.0
2017-10-14T16:23:35.323Z
Then is not the title of the question not appropriate?
luchonacho
CC BY-SA 3.0
2017-10-14T18:27:55.577Z
Thanks for alerting me to that, I've updated it. I had a change of mind while typing in the post, also, as you may have noticed my question text different from the one I posted originally. So, a lot of things going on in my mind, perhaps, stimulated by your questioning and comments above. So, what would you suggest? I derived the (annual) technology residuals and I am trying to compare them with factor prices, factor ratios and factor shares to evaluate and and pin down the bias and its sources. Just need a coherent strategy and a bit of motivation to start off.
london
CC BY-SA 3.0
2017-10-16T10:27:56.930Z
[This paper](https://ideas.repec.org/p/deg/conpap/c009_030.html) might be of help.
luchonacho
<macroeconomics><economic-growth><directed-technical-change>
How to pin down the bias and its/their sources in derived technical change indices?
london
# How to pin down the bias and its/their sources in derived technical change indices?

(Asked by: username_0 on 2017-10-13)

Using the following CES function I backed out factor-augmenting indices $A\_1$ and $A\_2$ using OECD country level macro data. This process is akin to Solow's (1956) method for deriving technology residuals. I want to examine the bias in the direction of technical change by examining the temporal evolution of the factor-augmenting indices.

$$Y=[((A\_1L^\alpha) K^{(1-\alpha)})^\sigma+ A\_2E^\sigma]^{1/\sigma}$$

The elasticity of substitution parameter $\sigma$ is 0.1 (Estimated using MLE). $Y$ is GDP. $E$ is energy flow.

There is no evidence of trend in the two technical change indices until the last few decades of the sample. From there on, the labour-augmenting index grows rapidly. The other index augmenting energy declines gradually. This divergence in the indices coincides with the sudden energy price shocks.

It might look pretty obvious that there was a labour saving and energy using bias in technical change. Though, I am stuck at this point, ***how would I link the derived factor-augmenting indices with factor prices and factor ratios to impart the technical change biases in a systematic way?***

---

I tried a slightly different formulation because of the nesting of the Cobb-Douglas function in CES. The evidence pointed out by `@username_1`
in previous posts shows that Cobb-Douglas is not an apropriate fit to the data given that there is little potential for $K$ and $L$ to be perfect substitutes.

An alternative formulation could be
$$Y=[(A\_lL^\nu+ A\_kK^\nu)^\frac{\sigma}{\nu}+ A\_eE^\sigma]^{1/\sigma}$$
However, estimating $\nu$ and $\sigma$ would not be easy due to the nonlinearities in the relationship. Thus, the first formulation was adopted. Any suggestions welcome! Alternative formulation, methods etc.

**Tags:** macroeconomics, economic-growth, directed-technical-change

> **Comments:**
> **username_1**: Interesting research there. Some comments/questions: (i) what is your estimate of $\sigma$? The interpretation of labour-saving/augmenting depends on its value. (ii) have you tested for changes in $\sigma$? (iii) ideally you want your production function to be [normalised](https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf?9fcd5fddf87498417164716f2bbee9fd). (iv) why assuming elasticity of substitution between capital and labour equal to one? Most of the evidence I have seen point toward $\sigma_{K,L}<1$. Maybe use a translog production function. (v) can you clarify your question?  
> **username_0**: @username_1 Many thanks, I've edited the post. Is it good enough?  
> **username_1**: 0.01? Really? So basically, no substitution between K-L bundle and energy? is this in line with other research? As far as I know the translog allows for quite a lot of freedom in terms of substitution. You do not face the nonlinearity problem.  
> **username_0**: Sorry, I hastily typed the original edit and ended up making a number of typos due to having to step out of my office. I've edited the post again. The elasticiy value is the average for several countries. Please have a look.  
> **username_1**: What is Y in your data? Notice value added might not be a good proxy for Y, because this measures contribution from non-contemporaneously produced factors of production and not from intermediate inputs like energy, which are also produced by K and L. This is, $E=f(K_E,L_E)$, both of which are included in $K$ and $L$.  
> **username_0**: It is GDP. Yes, I am aware of this and this exercise is my inital attempt at working out a strategy for pinning down the bias in technical change. I think, this will be a limitation for this work.The production function is not normalised either. So, will work on it for a few months. I will try translog function now. Also, updated the post.  
> **username_1**: I still do not understand the question. What else do you want to estimate? Why are your estimations of $A_i$ not sufficient for your interest?  
> **username_0**: Please note that $A_i$s are time varying indices, I should have typed them as $A_{i,t}$, my bad. Surely, the indices show the turns and twists that indicate possible bias, but I want to pin down where the bias stems from. For now, I am looking at the correlation between $A_{i,t}$s and energy price and wages. There is a positive correlation between $A_{e,t}$ and energy prices indicating energy saving-bias induced by energy shocks. But not sure if this approach is sufficient to make a conclusion. Also, please see the question in the post highlighted in bold.  
> **username_1**: So you do not want to pin down the bias but the source of the bias?  
> **username_0**: Yes, as I note in the last comment I can decipher the bias from the trend breaks in technology residuals, but also want to identify the sources of the bias. Though, I do not have clear strategy for this - just looking at various approaches. Any suggestions welcome.  
> **username_1**: Then is not the title of the question not appropriate?  
> **username_0**: Thanks for alerting me to that, I've updated it. I had a change of mind while typing in the post, also, as you may have noticed my question text different from the one I posted originally. So, a lot of things going on in my mind, perhaps, stimulated by your questioning and comments above. So, what would you suggest? I derived the (annual) technology residuals and I am trying to compare them with factor prices, factor ratios and factor shares to evaluate and and pin down the bias and its sources. Just need a coherent strategy and a bit of motivation to start off.  
> **username_1**: [This paper](https://ideas.repec.org/p/deg/conpap/c009_030.html) might be of help.  

---

## 0 answers

