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Cobb-Douglas production function with expenditures rather than units

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4

1

$begingroup$

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?

microeconomics econometrics applied-econometrics

share|improve this question

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

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add a comment | 

4

1

$begingroup$

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?

microeconomics econometrics applied-econometrics

share|improve this question

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

$endgroup$

add a comment | 

$begingroup$

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?

microeconomics econometrics applied-econometrics

share|improve this question

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

$endgroup$

share|improve this question

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

share|improve this question

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

asked 8 hours ago

leecarvalloleecarvallo

5833 bronze badges

2 Answers
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4

$begingroup$

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 to check if your function is CES instead of just a simple cobb-douglas.

share|improve this answer

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right?
  $endgroup$
  – leecarvallo
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @leecarvallo yep its different, see this here for more details. economics.stackexchange.com/questions/26144/… be sure to accept and upvote this answer if it answers your question. It helps move the site out of beta.
  $endgroup$
  – EconJohn♦
  6 hours ago
  
  
  

  
  
  
  

add a comment | 

2

$begingroup$

If prices are constant than 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 $a$ this way.

share|improve this answer

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

$endgroup$

add a comment | 

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4

$begingroup$

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 to check if your function is CES instead of just a simple cobb-douglas.

share|improve this answer

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right?
  $endgroup$
  – leecarvallo
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @leecarvallo yep its different, see this here for more details. economics.stackexchange.com/questions/26144/… be sure to accept and upvote this answer if it answers your question. It helps move the site out of beta.
  $endgroup$
  – EconJohn♦
  6 hours ago
  
  
  

  
  
  
  

add a comment | 

4

$begingroup$

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 to check if your function is CES instead of just a simple cobb-douglas.

share|improve this answer

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, thanks! And thanks for the recommendation! The translog production function is distinct from just log transforming a regular Cobb-Douglas production function, right?
  $endgroup$
  – leecarvallo
  6 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @leecarvallo yep its different, see this here for more details. economics.stackexchange.com/questions/26144/… be sure to accept and upvote this answer if it answers your question. It helps move the site out of beta.
  $endgroup$
  – EconJohn♦
  6 hours ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

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 to check if your function is CES instead of just a simple cobb-douglas.

share|improve this answer

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

$endgroup$

share|improve this answer

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

answered 6 hours ago

EconJohn♦EconJohn

3,54322 gold badges1111 silver badges4141 bronze badges

2

$begingroup$

If prices are constant than 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 $a$ this way.

share|improve this answer

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

$endgroup$

add a comment | 

2

$begingroup$

If prices are constant than 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 $a$ this way.

share|improve this answer

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

$endgroup$

add a comment | 

$begingroup$

If prices are constant than 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 $a$ this way.

share|improve this answer

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

$endgroup$

share|improve this answer

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges

answered 5 hours ago

BKayBKay

12.6k22 gold badges2424 silver badges6161 bronze badges