<blockquote>
<p>(1) why is imposing CES so important in our models?</p>
</blockquote>
<p>Because although its relatively quite general (relatively to some other widely used production functions like Cobb-Douglas - which is a special case of CES) it is still easy to estimate with parametric models and generally CES production functions are easy to work with (<a href="https://academic.oup.com/restud/article-abstract/30/2/73/1517685?redirectedFrom=fulltext" rel="nofollow noreferrer">McFadden 1963</a>).</p>
<p>Until very recently you need Cobb-Douglas, or some CES, with its unitary elasticity of substitution due to the normalization problem which precluded people from applying even more general form. For example, as discussed in <a href="https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf" rel="nofollow noreferrer">Klump et al (2011)</a> [emphasis mine]:</p>
<blockquote>
<p>Until recently, the application of production functions
with non-unitary substitution elasticities (i.e., non Cobb Douglas) was hampered
by empirical and theoretical uncertainties. As has recently been revealed, “normalization” of production functions and production-technology systems holds out
the promise of resolving many of those uncertainties and allowing considerations as
the role of the substitution elasticity and biased technical change to play a deeper
role in growth and business-cycle analysis. Normalization essentially implies representing the production function in consistent indexed number form. <strong>Without
normalization, it can be shown that the production function parameters have no
economic interpretation since they are dependent on the normalization point and
the elasticity of substitution itself</strong>. This feature significantly undermines estimation and comparative-static exercises, among other things.</p>
</blockquote>
<p>The above issue leads to bias in estimation (especially in parametric models) so its quite a serious issue. This is because, we can only use observable data for estimation but capital and labor are measured in completely different units (aside from the problem that we actually have no way of accurately measuring capital in the first place).  CES or Cobb-Douglass with unitary fixed elasticity gets around the issue by the virtue of elasticity of substitution being 1, and because the differences in units get absorbed into the scaling constant.</p>
<p>But what even more, as the above cited paper discussed, the problem of normalization was essentially solved for CES even with non-unitary elasticities making it even so more desirable function to use.  This is quite important since empirically elasticity of substitution is generally below unity (e.g. have a look at <a href="https://www.sciencedirect.com/science/article/abs/pii/S0047272799000249?casa_token=2-aYd3UIB1oAAAAA:S_dVDrAU0IptuS1-HZRgyVQ2ila5vbSTiw5ZTF_erpUFvZol2xmJdY9Qvmt-q_mXxQzXdp74r5Q" rel="nofollow noreferrer">Chirinko et al. 1999</a>, <a href="https://direct.mit.edu/rest/article-abstract/89/1/183/57638/Factor-Substitution-and-Factor-Augmenting" rel="nofollow noreferrer">Klump et al. 2007</a>, <a href="https://www.aeaweb.org/articles?id=10.1257/aer.100.4.1330" rel="nofollow noreferrer">Leon-Ledesma et al. 2010</a>).</p>
<p>Lastly, estimation of  production functions is riddled beyond belief with endogeneity issues. One way how to solve the endogeneity issues is to look to theory for guidance of how to set up model (especially how to properly specify the error term) to avoid these issues  (e.g. see the work of <a href="https://web.archive.org/web/20211112161908/https://www.jstor.org/stable/2171831?seq=1#page_scan_tab_contents" rel="nofollow noreferrer">Olley Pakes 1996</a>,  <a href="https://www.nber.org/papers/w7819" rel="nofollow noreferrer">Levinsohn-Petrin 2000</a> or <a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA13408?casa_token=YITxPu4O2wIAAAAA:_unIEQBPoYb8iN9nOKznktKIty_F4MYARvYO2UkwTKD93PSMR3jUOWt6PQqOreVP4MYenrWaUP4F--o" rel="nofollow noreferrer">Ackerberg, Caves and Frazer 2015</a>). But lot of these theoretical results are derived assuming CES production function, so you can't just use theoretical result derived assuming CES production to model the structure of the error term and nothing else.</p>
<blockquote>
<ol start="2">
<li>are there any serious papers or methods that allow for a non-CES production function?</li>
</ol>
</blockquote>
<p>Well yes, there are a lot of papers that apply translog production function which allows elasticity of substitution to change (<a href="https://link.springer.com/chapter/10.1007/978-1-349-18721-8_6" rel="nofollow noreferrer">Heathfield, Wibe 1987</a>). If you just put <a href="https://scholar.google.com/scholar?hl=en&amp;as_sdt=0,5&amp;q=translog%20production%20function%20estimation" rel="nofollow noreferrer">translog production function estimation into google scholar</a> you will get a lot of examples.</p>
<p>However, estimating translog production function parametrically can be problematic because relative to CES or Cobb-Douglass you have to estimate quite a lot of parameters to make it work. Especially if you want to include a lot of factors of production the number of parameters virtually explodes (Pavelescu, 2011), and this is a problem given the tendency of including more and more factors in recent years (e.g. materials, different types of capital and so on).</p>

CC BY-SA 4.0
2021-08-01T22:45:46.373Z
Only qualm is you cite OP and LP but not ACF. Great answer!
Michael Gmeiner
CC BY-SA 4.0
2021-08-01T22:49:26.907Z
@MichaelGmeiner thanks, I already fixed that, you are right the ACF paper is actually even better reference than OP and LP since they explain everything very clearly
1muflon1
CC BY-SA 4.0
2021-08-02T11:42:41.813Z
I have a followup question. I consider Gandhi, Navarro, Rivers (2020) to be an excellent paper. They allow for a fully nonparametric production function (i.e. it could include non-CES), and I see no mention of the normalization. I don't fully understand the normalization, could you explain why GNR don't need to worry about it?
Michael Gmeiner
CC BY-SA 4.0
2021-08-02T12:13:35.363Z
@MichaelGmeiner thats a good question. I do not have a ready answer for it of top of my hat because in the past and present I primarily work with the standard parametric models (e.g. OP, LP, ACF, Wooldridge 2009), because there is still so much you can get out of these models, and I primarily try to apply this to policy economics e.g. how different policies affect firm-level productivity, where it is preferable to use well tested methods for estimating productivity. If I would have to make an educated guess, I would say this is either because they have some other way how to deal with it
1muflon1
CC BY-SA 4.0
2021-08-02T12:16:23.300Z
they just don't stress it or it is because maybe in non-parametric models this is not an issue to begin with (I must admit I am not really an expert on non-parametric estimation),  but with parametric models the  lack of normalization creates problem because the difference in units (labor vs capital) loads on the estimated coefficient values thus biasing them, and in a such way the coefficients become meaningless. I guess, depending on how you estimate the non-parametric model perhaps that is not an issue to begin with? Consider, asking this as a separate question maybe someone knows the A
1muflon1
CC BY-SA 4.0
2021-08-02T13:14:10.053Z
Thank you, I see you're a graduate student. I'll be an AP in starting in the fall. Feel free to find me on LinkedIn/twitter and let me know if I can ever help you with anything.
Michael Gmeiner
CC BY-SA 4.0
2021-08-02T13:16:57.857Z
@MichaelGmeiner thanks for the offer I will keep that in the mind :)
1muflon1
CC BY-SA 4.0
2021-08-14T20:26:19.557Z
@1mulflon1 I think I figured this out about GNR. In a manner similar to Cobb-Douglas they have all variables logged, it just ends up being a big polynomial of inputs on the right side, and thus units get absorbed into the constant.
Michael Gmeiner
CC BY-SA 4.0
2021-08-01T19:16:00.550Z
2022-07-03T14:14:27.987Z
2022-07-03T14:14:27.987Z
1muflon1
<p>I know CES production functions dominate economics, but I was curious, why? I've never seen a research paper or presentation utilize any form of a production function that is not CES.</p>
<p>My question is twofold (1) why is imposing CES so important in our models? (2) are there any serious papers or methods that allow for a non-CES production function?</p>
<p>EDIT: I know <a href="https://www.journals.uchicago.edu/doi/full/10.1086/707736?af=R&amp;mobileUi=0" rel="nofollow noreferrer">Gandhi, Navarro, Rivers (2020)</a> allows for a fully nonparametric production function, I'm looking for papers that specifically consider the possibility of a non-CES production function.</p>

<production-function><ces-function>
Non CES Production Functions
Michael Gmeiner
# Non CES Production Functions

(Asked by: username_0 on 2021-08-01)

I know CES production functions dominate economics, but I was curious, why? I've never seen a research paper or presentation utilize any form of a production function that is not CES.

My question is twofold (1) why is imposing CES so important in our models? (2) are there any serious papers or methods that allow for a non-CES production function?

EDIT: I know [Gandhi, Navarro, Rivers (2020)](https://www.journals.uchicago.edu/doi/full/10.1086/707736?af=R&mobileUi=0) allows for a fully nonparametric production function, I'm looking for papers that specifically consider the possibility of a non-CES production function.

**Tags:** production-function, ces-function


---

## 1 answers

### Answer by username_1 on 2021-08-01. Score: 3 | Accepted answer

> (1) why is imposing CES so important in our models?

Because although its relatively quite general (relatively to some other widely used production functions like Cobb-Douglas - which is a special case of CES) it is still easy to estimate with parametric models and generally CES production functions are easy to work with ([McFadden 1963](https://academic.oup.com/restud/article-abstract/30/2/73/1517685?redirectedFrom=fulltext)).

Until very recently you need Cobb-Douglas, or some CES, with its unitary elasticity of substitution due to the normalization problem which precluded people from applying even more general form. For example, as discussed in [Klump et al (2011)](https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1294.pdf) [emphasis mine]:

> Until recently, the application of production functions
> with non-unitary substitution elasticities (i.e., non Cobb Douglas) was hampered
> by empirical and theoretical uncertainties. As has recently been revealed, “normalization” of production functions and production-technology systems holds out
> the promise of resolving many of those uncertainties and allowing considerations as
> the role of the substitution elasticity and biased technical change to play a deeper
> role in growth and business-cycle analysis. Normalization essentially implies representing the production function in consistent indexed number form. **Without
> normalization, it can be shown that the production function parameters have no
> economic interpretation since they are dependent on the normalization point and
> the elasticity of substitution itself**. This feature significantly undermines estimation and comparative-static exercises, among other things.

The above issue leads to bias in estimation (especially in parametric models) so its quite a serious issue. This is because, we can only use observable data for estimation but capital and labor are measured in completely different units (aside from the problem that we actually have no way of accurately measuring capital in the first place). CES or Cobb-Douglass with unitary fixed elasticity gets around the issue by the virtue of elasticity of substitution being 1, and because the differences in units get absorbed into the scaling constant.

But what even more, as the above cited paper discussed, the problem of normalization was essentially solved for CES even with non-unitary elasticities making it even so more desirable function to use. This is quite important since empirically elasticity of substitution is generally below unity (e.g. have a look at [Chirinko et al. 1999](https://www.sciencedirect.com/science/article/abs/pii/S0047272799000249?casa_token=2-aYd3UIB1oAAAAA:S_dVDrAU0IptuS1-HZRgyVQ2ila5vbSTiw5ZTF_erpUFvZol2xmJdY9Qvmt-q_mXxQzXdp74r5Q), [Klump et al. 2007](https://direct.mit.edu/rest/article-abstract/89/1/183/57638/Factor-Substitution-and-Factor-Augmenting), [Leon-Ledesma et al. 2010](https://www.aeaweb.org/articles?id=10.1257/aer.100.4.1330)).

Lastly, estimation of production functions is riddled beyond belief with endogeneity issues. One way how to solve the endogeneity issues is to look to theory for guidance of how to set up model (especially how to properly specify the error term) to avoid these issues (e.g. see the work of [Olley Pakes 1996](https://web.archive.org/web/20211112161908/https://www.jstor.org/stable/2171831?seq=1#page_scan_tab_contents), [Levinsohn-Petrin 2000](https://www.nber.org/papers/w7819) or [Ackerberg, Caves and Frazer 2015](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA13408?casa_token=YITxPu4O2wIAAAAA:_unIEQBPoYb8iN9nOKznktKIty_F4MYARvYO2UkwTKD93PSMR3jUOWt6PQqOreVP4MYenrWaUP4F--o)). But lot of these theoretical results are derived assuming CES production function, so you can't just use theoretical result derived assuming CES production to model the structure of the error term and nothing else.

> 2. are there any serious papers or methods that allow for a non-CES production function?

Well yes, there are a lot of papers that apply translog production function which allows elasticity of substitution to change ([Heathfield, Wibe 1987](https://link.springer.com/chapter/10.1007/978-1-349-18721-8_6)). If you just put [translog production function estimation into google scholar](https://scholar.google.com/scholar?hl=en&as_sdt=0,5&q=translog%20production%20function%20estimation) you will get a lot of examples.

However, estimating translog production function parametrically can be problematic because relative to CES or Cobb-Douglass you have to estimate quite a lot of parameters to make it work. Especially if you want to include a lot of factors of production the number of parameters virtually explodes (Pavelescu, 2011), and this is a problem given the tendency of including more and more factors in recent years (e.g. materials, different types of capital and so on).

> **Comments:**
> **username_0**: Only qualm is you cite OP and LP but not ACF. Great answer!  
> **username_1**: @MichaelGmeiner thanks, I already fixed that, you are right the ACF paper is actually even better reference than OP and LP since they explain everything very clearly  
> **username_0**: I have a followup question. I consider Gandhi, Navarro, Rivers (2020) to be an excellent paper. They allow for a fully nonparametric production function (i.e. it could include non-CES), and I see no mention of the normalization. I don't fully understand the normalization, could you explain why GNR don't need to worry about it?  
> **username_1**: @MichaelGmeiner thats a good question. I do not have a ready answer for it of top of my hat because in the past and present I primarily work with the standard parametric models (e.g. OP, LP, ACF, Wooldridge 2009), because there is still so much you can get out of these models, and I primarily try to apply this to policy economics e.g. how different policies affect firm-level productivity, where it is preferable to use well tested methods for estimating productivity. If I would have to make an educated guess, I would say this is either because they have some other way how to deal with it  
> **username_1**: they just don't stress it or it is because maybe in non-parametric models this is not an issue to begin with (I must admit I am not really an expert on non-parametric estimation),  but with parametric models the  lack of normalization creates problem because the difference in units (labor vs capital) loads on the estimated coefficient values thus biasing them, and in a such way the coefficients become meaningless. I guess, depending on how you estimate the non-parametric model perhaps that is not an issue to begin with? Consider, asking this as a separate question maybe someone knows the A  
> **username_0**: Thank you, I see you're a graduate student. I'll be an AP in starting in the fall. Feel free to find me on LinkedIn/twitter and let me know if I can ever help you with anything.  
> **username_1**: @MichaelGmeiner thanks for the offer I will keep that in the mind :)  
> **username_0**: @1mulflon1 I think I figured this out about GNR. In a manner similar to Cobb-Douglas they have all variables logged, it just ends up being a big polynomial of inputs on the right side, and thus units get absorbed into the constant.  

---

