<p>The simple answer to why the Cobb-Douglas functional form is used is because it is at least a log-linear approximation to some higher-order production function. That is, suppose you take a functional form that looks like this: <span class="math-container">$\log Y_t = f(A, K, L)$</span>. Then a <a href="https://en.wikipedia.org/wiki/Linear_approximation" rel="nofollow noreferrer">linear approximation</a> would look like the Cobb-Douglas production function. (For a small <span class="math-container">$1\%$</span> increase in <span class="math-container">$K$</span>, we get approximately an <span class="math-container">$\alpha \%$</span> increase in <span class="math-container">$Y$</span>.)</p>

<p>One potential source of the strange estimation results you're getting is likely a bias due to the <a href="https://en.wikipedia.org/wiki/Endogeneity_(econometrics)" rel="nofollow noreferrer">endogeneity of the regressors</a>. Total factor productivity (TFP) in your notation is <span class="math-container">$A$</span>. It is changing over time. In your regression, it shows up in the error term. Changes in productivity affect the composition of the inputs <span class="math-container">$L$</span> and <span class="math-container">$K$</span>. Thus, <span class="math-container">$L$</span> and <span class="math-container">$K$</span> are correlated with error term. To see this, solve any typical cost minimization for a firm. <span class="math-container">$L$</span> and <span class="math-container">$K$</span> will depend on <span class="math-container">$A$</span>. Also note that the behavior will be different in the short-run vs the long-run.</p>

<p>Hope this helps!</p>

<p><strong>EDIT:</strong></p>

<p>If you're interested, here is a <a href="https://github.com/jmbejara/comp-econ-sp19/blob/master/HW/hw-03-panel-data-methods/productivity-monte-carlo.ipynb" rel="nofollow noreferrer">problem set</a> (no solutions available) that walks you through a demonstration of the bias described above. In the end, one potential improvement to the estimation procedure is explored via Monte Carlo simulation. This improvement relies on the availability of panel data of firm input/output data.</p>

CC BY-SA 4.0
2019-09-28T14:43:57.513Z
thank you so much for your answer! When I look how TFP is calculated, it looks like it is calculated based on some production function (e.g. Cobb-Douglas) as A=Y/(L^α*K^β). Therefore it looks to me as a circular link, as we can either estimate α and β and then use them to calculate A, or we have to somehow know α and β without regression and then calculate A as a residual. What I am trying to understand is how to estimate/calibrate simple production function (e.g. Cobb-Douglas) without indirectly using α and β estimates (which are indirectly embedded in A) of other people. Thanks!
Petr
CC BY-SA 4.0
2019-09-28T22:18:41.110Z
@Petr The problem you are describing is a form of the [identification problem](https://en.wikipedia.org/wiki/Parameter_identification_problem) (or see the [identification](https://en.wikipedia.org/wiki/Identifiability) article). The problem isn't the functional form, it's that you don't have the data to estimate it. Unfortunately, there isn't an easy solution for you. You could add more assumption about the functional form of production, but this isn't likely a good solution. There is a whole literature devoted finding solutions to this problem, so you're not alone. Good luck!
jmbejara
CC BY-SA 4.0
2019-09-29T09:21:16.543Z
thanks for the clarification! I will further go through the links provided by you! And will reply later on if the question could be closed or if there would still be significant missing pieces of this picture in my head :)
Petr
CC BY-SA 4.0
2019-09-29T17:04:30.197Z
@Petr questions don't get closed on this site if they meet the standard. The way it works is that you should 'accept' the best answer by clicking on the tick below the voting buttons (which gives the answerer an extra 15 reputation points).
ahorn
CC BY-SA 4.0
2019-09-29T18:17:43.490Z
@ahorn, I will shortly accept the answer as soon as I understand that the answer solved my problem and before doing so I need to go through the links provided by jmbejara
Petr
CC BY-SA 4.0
2019-10-04T22:34:36.393Z
Ok, I see, without regression alpha and beta can be calculated from the condition of firm profit maximization by calculating derivatives of firm's profit by capital and by labor. Then after assigning these derivatives to zero, we can derive respective alpha and beta.
Petr
CC BY-SA 4.0
2019-10-04T22:56:51.023Z
@Petr I'm not sure what you mean, but it doesn't sound right. The bottom line is that you fundamentally can't get an unbiased estimate without additional data or assumptions.
jmbejara
CC BY-SA 4.0
2019-10-05T23:16:01.273Z
@jmbejara, I mean the following: (Step 1) Assume $Y = A*F(K,L)$. Then $dY/dt = Y/A*dA/dt + dF/K*dK/dt + dF/dL*dL/dt$. (Step 2): By differentiating firm Profit equation [$Profit = P*A*F(K,L) - wk*K - wl*L$] by labour and capital we have: $dF/dK = wk/P$ and $dF/dL = wl/P$. (Step 3): Substituting results from step 2 to results of step 1 we have: $dY/dt = Y/A*dA/dt + wk/P*dK/dt + wl/P*dL/dt$. After some arithmetical operations dA/A can be calculated as follows: $dA/A = dY/Y - wk*K/(P*Y)*dK/K - wl*L/(P*Y)*dL/L$ Conclusion: no Cobb-Douglas assumption is needed for growth accounting. Is this correct?
Petr
CC BY-SA 4.0
2019-10-05T23:29:35.980Z
@jmbejara, In addition to previous comment, looks like no regression is needed as all the parameters required for calculation of dA/A (e.g. Y, wl, P, L, K) are available in statistical data sources. It might be a bit tricky with wk, but I suppose it should be available or at least could be approximated. Therefore if we want to compare different geographical regions based on their TFP growth, can we calculate TFP growth based on the dA/A formula from my previous comment?
Petr
CC BY-SA 4.0
2019-10-06T18:16:01.943Z
Hi @Petr. Nice comment. BTW, I'd like to avoid a comment thread that is too long. And I think that the scope of the question seems to have shifted a bit in these comments. It might be nice at this point to open a new question for this. If you make the additional assumption that you have $w_k$ and $w_l$, things certainly do change. Good idea.
jmbejara
CC BY-SA 4.0
2020-04-18T09:47:30.763Z
@jmbejara Great answer! Unfortunately the link to the problem set is broken. Could you restore it? Thanks!
emeryville
CC BY-SA 4.0
2020-04-18T20:03:04.907Z
@emeryville Fixed!
jmbejara
CC BY-SA 4.0
2019-09-27T21:23:50.273Z
2020-04-18T20:02:52.637Z
2020-04-18T20:02:52.637Z
jmbejara
<p>I am trying to estimate a simple Cobb-Douglas production function <span class="math-container">$Y=AL^{\alpha}K^{\beta}$</span> 
on US data and get an unrealistic estimate of <span class="math-container">$\alpha$</span> and a high p-value for <span class="math-container">$\beta$</span>.
Structure of regression is as follows:</p>

<p><span class="math-container">$\frac 1 Y \frac{\mathrm dY}{\mathrm dt} = \frac 1 A \frac{\mathrm dA}{\mathrm dt} + \alpha \frac 1 L \frac{\mathrm dL}{\mathrm dt} 
+ \beta \frac 1 K \frac{\mathrm dK}{\mathrm dt}$</span></p>

<p>and assuming technological growth constant: </p>

<p><span class="math-container">$\text{growth}_Y = a_0 + \alpha\cdot \text{growth}_L + \beta\cdot \text{growth}_K$</span></p>

<p>I have the following output of linear regression in R:</p>

<p><span class="math-container">\begin{array}{lllll}
 &amp;\text{Estimate}\quad &amp; \text{Std. error}\quad &amp; \text{t value} &amp; \mathrm{Pr}(&gt;|t|) \\
\text{(Intercept)}\ &amp; 0.008256 &amp;   0.006070 &amp;  1.360  &amp;  0.186    \\
growth_L  &amp;  1.388552  &amp; 0.283867  &amp; 4.892 &amp; 5.47\mathrm e^{-05}\ ^{***} \\
growth_K   &amp; 0.194165   &amp; 0.255572  &amp; 0.760  &amp;  0.455 \\
\\
R^2 = 58\%
\end{array}</span></p>

<p>Estimate for the coefficient in front of <span class="math-container">$\text{growth}_L$</span> (<span class="math-container">$\hat\alpha$</span>) seems unrealistically high (1.388552) and estimate for coefficient for <span class="math-container">$\text{growth}_K$</span> (<span class="math-container">$\hat \beta$</span>) has high p-value (0.455).</p>

<p>Before running the regression I excluded outliers (observations outside 2.5% and 97.5% for <code>growth_gdp</code>, <code>growth_L</code> and <code>growth_K</code>).
I also checked correlation between <code>growth_L</code> and <code>growth_K</code> and it is 42% which is not that high to impact stability of coefficients of the regression that much.</p>

<p>Unfortunately I could not find a simple and reproducible analogue of the above analysis.</p>

<p>My questions are the following:</p>

<ol>
<li>Is there an empirical analysis of simple Cobb-Douglas function producing realistic estimates of coefficients? So that I would understand my key mistakes</li>
<li>If not, what would be the simplest form of production function which could be supported by empirical analysis?</li>
</ol>

<hr>

<p>P.S. For the full reproducibility of the analysis below is more info on data and R code used in estimation of the coefficients.
I take data from <a href="https://stats.oecd.org/viewhtml.aspx?datasetcode=EO105_INTERNET&amp;lang=en" rel="nofollow noreferrer">OECD Economic outlook database</a></p>

<p>Variables used: GDPV – states for Real GDP, HRS states for hours worked per worker, LF – states for number of workers in Economy, KTPV states for productive capital stock. I calculated L as <span class="math-container">$LF*HRS$</span></p>

<p>Full R code and the input data set can be found at <a href="https://pastebin.com/uqBm17SG" rel="nofollow noreferrer">pastebin</a>.</p>

CC BY-SA 4.0
2019-09-29T16:37:35.563Z
Thank you for this interesting question, Petr! [Here is advice for new users](https://economics.meta.stackexchange.com/q/164/175). For your information, you can share long code samples with https://pastebin.com/ .
ahorn
CC BY-SA 4.0
2019-09-29T18:08:43.623Z
Thanks for warm welcome ahorn! I transferred my code to pastebin and edited question accordingly.
Petr
<macroeconomics><econometrics><applied-econometrics><r>
Is there empirical evidence on usability of Cobb-Douglas production function?
Petr
# Is there empirical evidence on usability of Cobb-Douglas production function?

(Asked by: username_0 on 2019-09-26)

I am trying to estimate a simple Cobb-Douglas production function $Y=AL^{\alpha}K^{\beta}$
on US data and get an unrealistic estimate of $\alpha$ and a high p-value for $\beta$.
Structure of regression is as follows:

$\frac 1 Y \frac{\mathrm dY}{\mathrm dt} = \frac 1 A \frac{\mathrm dA}{\mathrm dt} + \alpha \frac 1 L \frac{\mathrm dL}{\mathrm dt}
+ \beta \frac 1 K \frac{\mathrm dK}{\mathrm dt}$

and assuming technological growth constant:

$\text{growth}\_Y = a\_0 + \alpha\cdot \text{growth}\_L + \beta\cdot \text{growth}\_K$

I have the following output of linear regression in R:

\begin{array}{lllll}
&\text{Estimate}\quad & \text{Std. error}\quad & \text{t value} & \mathrm{Pr}(>|t|) \\
\text{(Intercept)}\ & 0.008256 & 0.006070 & 1.360 & 0.186 \\
growth\_L & 1.388552 & 0.283867 & 4.892 & 5.47\mathrm e^{-05}\ ^{\*\*\*} \\
growth\_K & 0.194165 & 0.255572 & 0.760 & 0.455 \\
\\
R^2 = 58\%
\end{array}

Estimate for the coefficient in front of $\text{growth}\_L$ ($\hat\alpha$) seems unrealistically high (1.388552) and estimate for coefficient for $\text{growth}\_K$ ($\hat \beta$) has high p-value (0.455).

Before running the regression I excluded outliers (observations outside 2.5% and 97.5% for `growth_gdp`, `growth_L` and `growth_K`).
I also checked correlation between `growth_L` and `growth_K` and it is 42% which is not that high to impact stability of coefficients of the regression that much.

Unfortunately I could not find a simple and reproducible analogue of the above analysis.

My questions are the following:

1. Is there an empirical analysis of simple Cobb-Douglas function producing realistic estimates of coefficients? So that I would understand my key mistakes
2. If not, what would be the simplest form of production function which could be supported by empirical analysis?

---

P.S. For the full reproducibility of the analysis below is more info on data and R code used in estimation of the coefficients.
I take data from [OECD Economic outlook database](https://stats.oecd.org/viewhtml.aspx?datasetcode=EO105_INTERNET&lang=en)

Variables used: GDPV – states for Real GDP, HRS states for hours worked per worker, LF – states for number of workers in Economy, KTPV states for productive capital stock. I calculated L as $LF\*HRS$

Full R code and the input data set can be found at [pastebin](https://pastebin.com/uqBm17SG).

**Tags:** macroeconomics, econometrics, applied-econometrics, r
> **Comments:**
> **username_1**: Thank you for this interesting question, Petr! [Here is advice for new users](https://economics.meta.stackexchange.com/q/164/175). For your information, you can share long code samples with https://pastebin.com/ .  
> **username_0**: Thanks for warm welcome ahorn! I transferred my code to pastebin and edited question accordingly.  

---

## 1 answers

### Answer by username_2 on 2019-09-27. Score: 6 | Accepted answer

The simple answer to why the Cobb-Douglas functional form is used is because it is at least a log-linear approximation to some higher-order production function. That is, suppose you take a functional form that looks like this: $\log Y\_t = f(A, K, L)$. Then a [linear approximation](https://en.wikipedia.org/wiki/Linear_approximation) would look like the Cobb-Douglas production function. (For a small $1\%$ increase in $K$, we get approximately an $\alpha \%$ increase in $Y$.)

One potential source of the strange estimation results you're getting is likely a bias due to the [endogeneity of the regressors](https://en.wikipedia.org/wiki/Endogeneity_(econometrics)). Total factor productivity (TFP) in your notation is $A$. It is changing over time. In your regression, it shows up in the error term. Changes in productivity affect the composition of the inputs $L$ and $K$. Thus, $L$ and $K$ are correlated with error term. To see this, solve any typical cost minimization for a firm. $L$ and $K$ will depend on $A$. Also note that the behavior will be different in the short-run vs the long-run.

Hope this helps!

**EDIT:**

If you're interested, here is a [problem set](https://github.com/jmbejara/comp-econ-sp19/blob/master/HW/hw-03-panel-data-methods/productivity-monte-carlo.ipynb) (no solutions available) that walks you through a demonstration of the bias described above. In the end, one potential improvement to the estimation procedure is explored via Monte Carlo simulation. This improvement relies on the availability of panel data of firm input/output data.

> **Comments:**
> **username_0**: thank you so much for your answer! When I look how TFP is calculated, it looks like it is calculated based on some production function (e.g. Cobb-Douglas) as A=Y/(L^α*K^β). Therefore it looks to me as a circular link, as we can either estimate α and β and then use them to calculate A, or we have to somehow know α and β without regression and then calculate A as a residual. What I am trying to understand is how to estimate/calibrate simple production function (e.g. Cobb-Douglas) without indirectly using α and β estimates (which are indirectly embedded in A) of other people. Thanks!  
> **username_2**: @Petr The problem you are describing is a form of the [identification problem](https://en.wikipedia.org/wiki/Parameter_identification_problem) (or see the [identification](https://en.wikipedia.org/wiki/Identifiability) article). The problem isn't the functional form, it's that you don't have the data to estimate it. Unfortunately, there isn't an easy solution for you. You could add more assumption about the functional form of production, but this isn't likely a good solution. There is a whole literature devoted finding solutions to this problem, so you're not alone. Good luck!  
> **username_0**: thanks for the clarification! I will further go through the links provided by you! And will reply later on if the question could be closed or if there would still be significant missing pieces of this picture in my head :)  
> **username_1**: @Petr questions don't get closed on this site if they meet the standard. The way it works is that you should 'accept' the best answer by clicking on the tick below the voting buttons (which gives the answerer an extra 15 reputation points).  
> **username_0**: @ahorn, I will shortly accept the answer as soon as I understand that the answer solved my problem and before doing so I need to go through the links provided by jmbejara  
> **username_0**: Ok, I see, without regression alpha and beta can be calculated from the condition of firm profit maximization by calculating derivatives of firm's profit by capital and by labor. Then after assigning these derivatives to zero, we can derive respective alpha and beta.  
> **username_2**: @Petr I'm not sure what you mean, but it doesn't sound right. The bottom line is that you fundamentally can't get an unbiased estimate without additional data or assumptions.  
> **username_0**: @jmbejara, I mean the following: (Step 1) Assume $Y = A*F(K,L)$. Then $dY/dt = Y/A*dA/dt + dF/K*dK/dt + dF/dL*dL/dt$. (Step 2): By differentiating firm Profit equation [$Profit = P*A*F(K,L) - wk*K - wl*L$] by labour and capital we have: $dF/dK = wk/P$ and $dF/dL = wl/P$. (Step 3): Substituting results from step 2 to results of step 1 we have: $dY/dt = Y/A*dA/dt + wk/P*dK/dt + wl/P*dL/dt$. After some arithmetical operations dA/A can be calculated as follows: $dA/A = dY/Y - wk*K/(P*Y)*dK/K - wl*L/(P*Y)*dL/L$ Conclusion: no Cobb-Douglas assumption is needed for growth accounting. Is this correct?  
> **username_0**: @jmbejara, In addition to previous comment, looks like no regression is needed as all the parameters required for calculation of dA/A (e.g. Y, wl, P, L, K) are available in statistical data sources. It might be a bit tricky with wk, but I suppose it should be available or at least could be approximated. Therefore if we want to compare different geographical regions based on their TFP growth, can we calculate TFP growth based on the dA/A formula from my previous comment?  
> **username_2**: Hi @Petr. Nice comment. BTW, I'd like to avoid a comment thread that is too long. And I think that the scope of the question seems to have shifted a bit in these comments. It might be nice at this point to open a new question for this. If you make the additional assumption that you have $w_k$ and $w_l$, things certainly do change. Good idea.  
> **username_3**: @jmbejara Great answer! Unfortunately the link to the problem set is broken. Could you restore it? Thanks!  
> **username_2**: @emeryville Fixed!  

---

