<p>If prices are constant then quantities are proportional to expenditures. Consider :</p>

<p><span class="math-container">$$ Y=AK^{\alpha}L^{\beta} = A(\frac{E_{K}}{r})^{\alpha}(\frac{E_{L}}{w})^{\beta} $$</span>
<span class="math-container">$$ = (\frac{A}{r^\alpha w^\alpha})(E_{K})^{\alpha}(E_{L})^{\beta} $$</span>
<span class="math-container">$$ = \tilde{A}(E_{K})^{\alpha}(E_{L})^{\beta} $$</span></p>

<p>If prices don't vary too much this may be an acceptable approximation. However, notice that this is a log-additive function:
<span class="math-container">$$ \ln{Y_t} = y_t = a - \alpha \cdot r_t - \beta \cdot w_t + \alpha \cdot \ln (E_{K,t})  + \beta \cdot \ln (E_{L,t})$$</span></p>

<p>If you estimate a regression with time fixed effects, it absorbs the <span class="math-container">$a - \alpha \cdot r_t - \beta \cdot w_t $</span> term and your expenditures regressions give the same results for <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> as if you knew the quantities. If you want to know <span class="math-container">$\alpha$</span> or <span class="math-container">$\beta$</span> this is fine, but you won't identify <span class="math-container">$a$</span> / <span class="math-container">$A$</span> this way. </p>

CC BY-SA 4.0
2019-08-08T15:32:28.153Z
Thank you! Yes, I'm primarily interested in $\alpha$ and $\beta$. Out of curiosity, why do we need prices to be relatively constant in order to make this approximation?
leecarvallo
CC BY-SA 4.0
2019-08-08T16:05:34.600Z
Just because if $r$ and $w$ are constant than $\tilde{A}$ is a constant just like $A$ is in your specification. But if they are time varying then $\tilde{A}$ is also time varying. But you don't need to make that assumption to identify $\alpha$ and $\beta$, so it sounds like you are in luck.
BKay
CC BY-SA 4.0
2019-08-07T19:20:29.440Z
2019-08-08T16:26:09.913Z
2019-08-08T16:26:09.913Z
BKay
<p>Sure you can, just that your interpretation of your variables in your analysis changes however. In this case you are analyzing how investment in differing factors of production affect output.</p>

<p>I'd recommend that you may want to estimate a more flexible functional form like the <a href="https://www.youtube.com/watch?v=vKo9g5wr5G0" rel="noreferrer">Translog Production Function</a> to check if your function is CES instead of just a simple cobb-douglas.</p>

CC BY-SA 4.0
2019-08-07T18:49:24.563Z
Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right?
leecarvallo
CC BY-SA 4.0
2019-08-07T18:56:17.380Z
@leecarvallo yep its different, see this here for more details. https://economics.stackexchange.com/questions/26144/deriving-the-translog-production-function be sure to accept and upvote this answer if it answers your question. It helps move the site  out of beta.
EconJohn
CC BY-SA 4.0
2019-08-07T18:44:44.473Z
2019-08-07T18:44:44.473Z
EconJohn
<p>I want to estimate a standard Cobb-Douglas production function of the form </p>

<p><span class="math-container">$ Y=AK^{\alpha}L^{\beta}$</span>. However, I only have data on labour expenditures, not units of labour. I don't have data on the price of labour <span class="math-container">$w$</span> either, so I can't back out the units of labour from expenditures. Is it okay to estimate the production function (in a GMM regression framework) using labour expenditures instead of units of labour?</p>

<microeconomics><econometrics><applied-econometrics>
Cobb-Douglas production function with expenditures rather than units
leecarvallo
# Cobb-Douglas production function with expenditures rather than units

(Asked by: username_0 on 2019-08-07)

I want to estimate a standard Cobb-Douglas production function of the form

$ Y=AK^{\alpha}L^{\beta}$. However, I only have data on labour expenditures, not units of labour. I don't have data on the price of labour $w$ either, so I can't back out the units of labour from expenditures. Is it okay to estimate the production function (in a GMM regression framework) using labour expenditures instead of units of labour?

**Tags:** microeconomics, econometrics, applied-econometrics


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## 2 answers

### Answer by username_2 on 2019-08-07. Score: 9 | Accepted answer

Sure you can, just that your interpretation of your variables in your analysis changes however. In this case you are analyzing how investment in differing factors of production affect output.

I'd recommend that you may want to estimate a more flexible functional form like the [Translog Production Function](https://www.youtube.com/watch?v=vKo9g5wr5G0) to check if your function is CES instead of just a simple cobb-douglas.

> **Comments:**
> **username_0**: Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right?  
> **username_2**: @username_0 yep its different, see this here for more details. https://economics.stackexchange.com/questions/26144/deriving-the-translog-production-function be sure to accept and upvote this answer if it answers your question. It helps move the site  out of beta.  

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### Answer by username_1 on 2019-08-07. Score: 7

If prices are constant then quantities are proportional to expenditures. Consider :

$$ Y=AK^{\alpha}L^{\beta} = A(\frac{E\_{K}}{r})^{\alpha}(\frac{E\_{L}}{w})^{\beta} $$
$$ = (\frac{A}{r^\alpha w^\alpha})(E\_{K})^{\alpha}(E\_{L})^{\beta} $$
$$ = \tilde{A}(E\_{K})^{\alpha}(E\_{L})^{\beta} $$

If prices don't vary too much this may be an acceptable approximation. However, notice that this is a log-additive function:
$$ \ln{Y\_t} = y\_t = a - \alpha \cdot r\_t - \beta \cdot w\_t + \alpha \cdot \ln (E\_{K,t}) + \beta \cdot \ln (E\_{L,t})$$

If you estimate a regression with time fixed effects, it absorbs the $a - \alpha \cdot r\_t - \beta \cdot w\_t $ term and your expenditures regressions give the same results for $\alpha$ and $\beta$ as if you knew the quantities. If you want to know $\alpha$ or $\beta$ this is fine, but you won't identify $a$ / $A$ this way.

> **Comments:**
> **username_0**: Thank you! Yes, I'm primarily interested in $\alpha$ and $\beta$. Out of curiosity, why do we need prices to be relatively constant in order to make this approximation?  
> **username_1**: Just because if $r$ and $w$ are constant than $\tilde{A}$ is a constant just like $A$ is in your specification. But if they are time varying then $\tilde{A}$ is also time varying. But you don't need to make that assumption to identify $\alpha$ and $\beta$, so it sounds like you are in luck.  

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