<p>As per the previous answer (from EconJohn), the log-linear form of the production function then lends itself to regression analysis.  With the Cobb-Douglas form of a production function (i.e. the example you first cited) you can interpret the coefficients of the log-linear form as being measures of elasticity.</p>

<p>N.B. If your data is heteroskedastic, log-linearizing the production function will produce biased and inconsistent parameter estimates. This is because you are attempting to log-linearize a multiplicative error (mean 1) into an additive error (mean zero), yet the expectation of the mean of a random variable is not the same as the mean of the expectation by Jensen's inequality. </p>

CC BY-SA 3.0
2016-12-08T09:55:19.073Z
This is a comment from user Robert Brown which he tried to edit into the answer directly: «N.B. If your data is heteroskedastic, log-linearizing the production function will produce biased and inconsistent parameter estimates. This is because you are attempting to log-linearize a multiplicative error (mean 1) into an additive error (mean zero), yet the expectation of the mean of a random variable is not the same as the mean of the expectation by Jensen's inequality. »
An old man in the sea.
CC BY-SA 3.0
2016-12-06T20:05:32.120Z
2016-12-08T23:40:31.267Z
2016-12-08T23:40:31.267Z
B. Koch
<p>The following is the basic idea if we are to estimate the parameters by linear regression.</p>

<ol>
<li><p>Take the natural log of the production function <span class="math-container">$F(L,K)=L^aK^b$</span>, you will then get <span class="math-container">$$\ln(F)=a\ln(L)+b\ln(K).$$</span></p></li>
<li><p>For each entity (e.g., firm) <span class="math-container">$i$</span>, collect data on the production level <span class="math-container">$F_i$</span>, the amount of labour <span class="math-container">$L_i$</span>, and the amount of capital <span class="math-container">$K_i$</span>. Note that measurement issues may arise and that it is important to be explicit about how one is to interpret 'production', 'labour' and 'capital', and how to measure them.</p></li>
<li><p>Apply the transformation <span class="math-container">$X_i\mapsto \ln(X_i)$</span> for each variable <span class="math-container">$X_i$</span> for each entity <span class="math-container">$i$</span> in your dataset. (If there is any <span class="math-container">$X_i$</span> such that <span class="math-container">$X_i\leq 0$</span> then this will not work. You may delete those observations <span class="math-container">$X_i$</span> for which <span class="math-container">$X_i\leq 0$</span>. If it is real data, think about the following question. "Why is <span class="math-container">$X_i\leq 0$</span>? Has the dataset been wrongly registered?") Economists also apply the transformation <span class="math-container">$X_i\mapsto\ln(X_i+\epsilon)$</span> as an approximation when <span class="math-container">$X_i\geq0$</span> and <span class="math-container">$X_i=0$</span> for some <span class="math-container">$i$</span>, where <span class="math-container">$\epsilon&gt;0$</span> is some small number.</p></li>
<li><p>Now, regress <span class="math-container">$\ln(F)$</span> on <span class="math-container">$\ln(L)$</span> and <span class="math-container">$\ln(K)$</span> using your converted dataset (cf. 3) under the assumption that the intercept term is equal to <span class="math-container">$0$</span>. This gives you estimated values <span class="math-container">$\hat{a}$</span> and <span class="math-container">$\hat{b}$</span> of <span class="math-container">$a$</span> and <span class="math-container">$b$</span>. Hence, your estimated production function, which is supposed to capture some general tendency regarding production, is <span class="math-container">$$\hat{F}(L,K)=L^\hat{a}K^\hat{b}.$$</span></p></li>
<li><p>(If the production function is written as <span class="math-container">$F(K,L)=AL^aK^b$</span>, where <span class="math-container">$A$</span> captures technology for example, you will get an intercept term <span class="math-container">$\ln(A)$</span>.)</p></li>
</ol>

<p>If we are using <a href="https://en.wikipedia.org/wiki/Nonlinear_regression" rel="nofollow noreferrer">nonlinear regression</a>, then we take the nonlinear least squares (NLLS) estimate of <span class="math-container">$a$</span> and <span class="math-container">$b$</span> by solving <span class="math-container">$$\min_{a,\, b}\sum_i\big(F_i-L_i^aK_i^b)^2.$$</span> Numerical computing softwares (which uses algorithms such as the <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm" rel="nofollow noreferrer">Gauss-Newton algorithm</a>) may assist you when finding the NLLS estimate. </p>

CC BY-SA 3.0
2017-01-03T23:30:34.207Z
Note that interpretation of $\hat{a}$ and $\hat{b}$ might be different, since estimations shows how  $ln(F)$ will change if the $ln(L)$ will change
Yorgos
CC BY-SA 4.0
2016-12-06T20:52:41.213Z
2019-12-19T09:57:20.180Z
2019-12-19T09:57:20.180Z
Elias
<p>In a standard economics education we learn about production functions, indicating an output as a function of a given input of capital and labour.</p>

<p>An average model looks like this:</p>

<p><strong>(1)</strong>    $F(L,K)=L^{a}K^b$</p>

<p>When dealing with real data however, we are first exposed to regression models which are additive in nature which look like this:</p>

<p><strong>(2)</strong>      $y_i={\beta}_0+{\beta}_1x_1+{\beta}_2x_2+...+{\beta}_nx_n$</p>

<p>How do we get a production function that looks like (1) when dealing with real data? </p>

CC BY-SA 3.0
2016-12-06T18:15:55.260Z
I don't clearly understand the question: do you mean how do we get Eq. (1) from Eq (2)? Or is the question more general?
emeryville
CC BY-SA 3.0
2016-12-06T18:21:16.537Z
If you take the log of (1) you get $\ln(F)=a\ln(L)+b\ln(k)$, which has the linear additive form of (2) in its parameters.
Ubiquitous
CC BY-SA 3.0
2016-12-06T18:50:14.337Z
@Ubiquitous http://meta.economics.stackexchange.com/a/1651/1601
Giskard
<production-function><applied-econometrics>
How do we estimate production functions?
EconJohn
# How do we estimate production functions?

(Asked by: username_0 on 2016-12-06)

In a standard economics education we learn about production functions, indicating an output as a function of a given input of capital and labour.

An average model looks like this:

**(1)** $F(L,K)=L^{a}K^b$

When dealing with real data however, we are first exposed to regression models which are additive in nature which look like this:

**(2)** $y\_i={\beta}\_0+{\beta}\_1x\_1+{\beta}\_2x\_2+...+{\beta}\_nx\_n$

How do we get a production function that looks like (1) when dealing with real data?

**Tags:** production-function, applied-econometrics
> **Comments:**
> **username_1**: I don't clearly understand the question: do you mean how do we get Eq. (1) from Eq (2)? Or is the question more general?  
> **username_2**: If you take the log of (1) you get $\ln(F)=a\ln(L)+b\ln(k)$, which has the linear additive form of (2) in its parameters.  
> **username_3**: @Ubiquitous http://meta.economics.stackexchange.com/a/1651/1601  

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## 2 answers

### Answer by username_6 on 2016-12-06. Score: 8 | Accepted answer

The following is the basic idea if we are to estimate the parameters by linear regression.

1. Take the natural log of the production function $F(L,K)=L^aK^b$, you will then get $$\ln(F)=a\ln(L)+b\ln(K).$$
2. For each entity (e.g., firm) $i$, collect data on the production level $F\_i$, the amount of labour $L\_i$, and the amount of capital $K\_i$. Note that measurement issues may arise and that it is important to be explicit about how one is to interpret 'production', 'labour' and 'capital', and how to measure them.
3. Apply the transformation $X\_i\mapsto \ln(X\_i)$ for each variable $X\_i$ for each entity $i$ in your dataset. (If there is any $X\_i$ such that $X\_i\leq 0$ then this will not work. You may delete those observations $X\_i$ for which $X\_i\leq 0$. If it is real data, think about the following question. "Why is $X\_i\leq 0$? Has the dataset been wrongly registered?") Economists also apply the transformation $X\_i\mapsto\ln(X\_i+\epsilon)$ as an approximation when $X\_i\geq0$ and $X\_i=0$ for some $i$, where $\epsilon>0$ is some small number.
4. Now, regress $\ln(F)$ on $\ln(L)$ and $\ln(K)$ using your converted dataset (cf. 3) under the assumption that the intercept term is equal to $0$. This gives you estimated values $\hat{a}$ and $\hat{b}$ of $a$ and $b$. Hence, your estimated production function, which is supposed to capture some general tendency regarding production, is $$\hat{F}(L,K)=L^\hat{a}K^\hat{b}.$$
5. (If the production function is written as $F(K,L)=AL^aK^b$, where $A$ captures technology for example, you will get an intercept term $\ln(A)$.)

If we are using [nonlinear regression](https://en.wikipedia.org/wiki/Nonlinear_regression), then we take the nonlinear least squares (NLLS) estimate of $a$ and $b$ by solving $$\min\_{a,\, b}\sum\_i\big(F\_i-L\_i^aK\_i^b)^2.$$ Numerical computing softwares (which uses algorithms such as the [Gauss-Newton algorithm](https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm)) may assist you when finding the NLLS estimate.

> **Comments:**
> **username_7**: Note that interpretation of $\hat{a}$ and $\hat{b}$ might be different, since estimations shows how  $ln(F)$ will change if the $ln(L)$ will change  

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### Answer by username_4 on 2016-12-06. Score: 1

As per the previous answer (from EconJohn), the log-linear form of the production function then lends itself to regression analysis. With the Cobb-Douglas form of a production function (i.e. the example you first cited) you can interpret the coefficients of the log-linear form as being measures of elasticity.

N.B. If your data is heteroskedastic, log-linearizing the production function will produce biased and inconsistent parameter estimates. This is because you are attempting to log-linearize a multiplicative error (mean 1) into an additive error (mean zero), yet the expectation of the mean of a random variable is not the same as the mean of the expectation by Jensen's inequality.

> **Comments:**
> **username_5**: This is a comment from user Robert Brown which he tried to edit into the answer directly: «N.B. If your data is heteroskedastic, log-linearizing the production function will produce biased and inconsistent parameter estimates. This is because you are attempting to log-linearize a multiplicative error (mean 1) into an additive error (mean zero), yet the expectation of the mean of a random variable is not the same as the mean of the expectation by Jensen's inequality. »  

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