<p>According to the discussion in the seven page article available under this link (splash page):</p>
<p><a href="https://nsereview.org/index.php/NSER/article/view/79" rel="nofollow noreferrer">https://nsereview.org/index.php/NSER/article/view/79</a></p>
<blockquote>
<p>Using a Cobb-Douglas production function, regressions found a fairly good fit to the data and appeared to confirm the marginal productivity theory of distribution. While some were puzzled by the seemingly too-good-to-be-true  empirical  confirmation  of  a  theory  with  shaky  foundations,  Anwar Shaikh  demonstrated  forty years ago that “so long as aggregate shares are constant, an aggregate Cobb-Douglas function having apparently ‘constant returns to scale’ will always provide an exact fit, for any data whatsoever” and “will seem to also possess ‘marginal products  equal  to  respective  factor  rewards’”.  Any  data!  From  any  kind  of  economy, including a Humbug economy! The results are guaranteed not due to laws of economics but rather due to laws of algebra.</p>
</blockquote>
<blockquote>
<p>Robert Solow replied to Anwar as follows: The  factor-share  device  of  my  1957  article  is  in  no  sense  a test of  aggregate production  functions  or  marginal  productivity  or  anything  else.  It  merely  shows how  one  goes  about  interpreting  given  time  series  if  one  starts  by assuming that they were generated from a production function and that the competitive marginal-product relations apply. [Solow 1974: 121, emphasis in original - Solow, R.M.1974, “Law of production and laws of algebra: the Humbug production function: A comment”, Review of Economics and Statistics, 56(1).]</p>
</blockquote>
<p>The aggregate production function models are not considered to be empirical measures. These models make it appear as though aggregate income is or could be a function of the marginal contribution of labor and capital. The article states that such models can fit a curve to any data which could have been generated by an aggregate production function with the specified parameters. The post-keynesian argument is that there is no singular production process, there is no empirical measure of aggregate capital, and there is no deterministic model for the historical allocation of aggregate income to shares of labor and profit. Instead there is a complex interaction among economic and political agents, an outcome, and curve fitting exercise called the production function.</p>

CC BY-SA 4.0
2021-07-24T14:47:41.127Z
You seem to be commenting on the [other answer](https://economics.stackexchange.com/a/45954/1601), not trying to answer the question itself.
Giskard
CC BY-SA 4.0
2021-07-24T14:55:58.120Z
The question includes this comment "I heard that this is possible in neoclassical economics, but I'm not sure how." The "how" in general is to assume some production function Y = f(K,L) where income is a function of the contribution of capital and the contribution of labor. The meaning of these functions is the subject of debate in the field of economics. My answer relates to the meaning of the assumptions applied in those methods which is more important to understand in terms of economic reasoning than the exact algebraic method.
SystemTheory
CC BY-SA 4.0
2021-07-24T15:19:11.067Z
I see; alright.
Giskard
CC BY-SA 4.0
2021-07-30T15:35:13.970Z
this does not answer the question, the source and quote talks about aggregate production function, but the question ask about micro level individual production function
csilvia
CC BY-SA 4.0
2021-07-30T20:18:14.470Z
Neither micro nor macro production functions are empirical measurements because there is no objective way to measure of the contribution of capital compared to the contribution of labor. Production functions should be recognized as curve fitting models and with no deterministic model for describing the empirical (measured) contribution of capital and the empirical (measured) contribution of labor to the output produced.
SystemTheory
CC BY-SA 4.0
2021-07-24T14:04:30.453Z
2021-07-24T15:46:07.697Z
2021-07-24T15:46:07.697Z
SystemTheory
<p>You can do this in various ways for a <em>value of output</em> (e.g. goods and services) firm produces but not really very reliably for value of firm itself.</p>
<p><strong>Value of Output</strong></p>
<p>When it comes to value of output then this boils down to estimating production function of the company. For example, assuming that company has Cobb-Douglas production function:</p>
<p><span class="math-container">$$Y = AK^\beta L^{(1-\beta)}$$</span></p>
<p>where <span class="math-container">$Y$</span> is the value of output, <span class="math-container">$A$</span> is the multifactor productivity, <span class="math-container">$K$</span> is the capital, and <span class="math-container">$L$</span> labor input respectively. You can just log-linearize the expression (below I am using common notation for logs in economics <span class="math-container">$x= \ln X$</span>) and turn it into</p>
<p><span class="math-container">$$ y = a + \beta k + (1-\beta) l $$</span></p>
<p>this can be then set up as a panel regression model regression model (where <span class="math-container">$a$</span> is decomposed to <span class="math-container">$\beta_0$</span>, <span class="math-container">$\omega_{it}$</span> and <span class="math-container">$\epsilon_{it}$</span>:</p>
<p><span class="math-container">$$ y = \beta_0 + \omega_{it} + \beta_1 k_{it} + \beta_2 l_{it} + \epsilon_{it}$$</span></p>
<p>However, you cannot just use naïve panel regression as capital is incredibly difficult to properly measure, so you will likely be only able to find accounting data which are created for a purpose of minimizing firm's tax obligations not to accurately describe capital stock of company or its input utilization.</p>
<p>Thankfully, over the years various methodologies and empirical strategies were developed to overcome this issue (although it is not 100% solved problem). The main solutions to this problem are the so called <a href="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> and <a href="https://www.sciencedirect.com/science/article/abs/pii/S0165176509001487?via%3Dihub" rel="nofollow noreferrer">Wooldridge (2009)</a> models (all are variants on the general method of moments [GMM]). These models use various instruments to correct for the endogeneity issue caused by capital measurement errors (although again it's not 100% solved issue but the above methods are state of the art).</p>
<p><strong>Firm Value</strong></p>
<p>When it comes to value of firm, that is trickier. From an economic perspective value of firm depends almost solely on expectations of its future profitability and streams of cash it creates through dividends paid from the said profit (see discussion in Mishkin and Eakins Financial Markets and Institutions in the chapter on stock market). Firms like Nokia or Blockbuster or Kodak had both large capital stocks and number of employees even at a point where firm value was driven to near zero.</p>
<p>Even though profitability on some level depends use of capital and labor the relationship is much less clear cut when it comes to firm value as when it comes to firm output.  In addition you have the same capital (mis)measurement problem as in the previous section, but now with even less good models to suggest appropriate instruments to correct for it.</p>
<p>However, there were recent attempts at doing this. For example, <a href="https://www.nber.org/system/files/working_papers/w26112/w26112.pdf" rel="nofollow noreferrer">Belo et al (2019)</a> have recent working paper on this topic. They estimate model based on following firm valuation ratio:</p>
<p><span class="math-container">$$VR_{it} = q^P_{it}\frac{K^P_{it+1}}{A_{it+1}} + q^L_{it}\frac{L_{it+1}}{A_{it+1}}+ q^K_{it}\frac{K^K_{it+1}}{A_{it+1}} + q^B_{it}\frac{K^B_{it+1}}{A_{it+1}} \tag{*}$$</span></p>
<p>where <span class="math-container">$VR$</span> is the firm valuation ratio (<span class="math-container">$V R_{it} \equiv (P_{it} + B_{it+1}) /A_{it+1}$</span>), where <span class="math-container">$P$</span> is the firm's stock price, <span class="math-container">$B$</span> is the firm level debt, and <span class="math-container">$A$</span> is the measure of firm effective assets. Next, <span class="math-container">$q^j, j=P,K,B,L$</span> are shadow prices for the firm price, two types of capital stocks (physical and brand capital <span class="math-container">$K^K, K^B$</span>) and labor which can be inferred from firm level data, K are various measures of capital (depending on superscript) and <span class="math-container">$L$</span> is labor stock. They estimate the model parameters by minimizing the squared distance between the portfolio-level
observed and model-implied valuation ratio moments at each point in time.</p>
<p>That is they first set up the relationship between observed valuation ratios given by:</p>
<p><span class="math-container">$$VR^{XSM}_{jt} (\Theta) = \sum_i \frac{\hat{V R}_{it}}{N_{jt}}, i \in \text{ portfolio } j$$</span>,</p>
<p>where the left hand side has observed valuations (that depend on parameters <span class="math-container">$\Theta$</span> that are parameters of the model given by (*), and right hand side valuations to be estimated, which are later estimated setting up:</p>
<p><span class="math-container">$$ VR^{XSM}_{jt}= \hat{VR}^{XSM}_{jt} (\Theta) + \epsilon_{it}$$</span></p>
<p>where <span class="math-container">$\epsilon$</span> captures measurement error in the portfolio moments, and the parameters are in the end estimated using:</p>
<p><span class="math-container">$$\hat{Θ }= \arg \min_{\Theta} \sum_{t=1}^T  \sum_{n=1}^N \left( VR^{XSM}_{jt}-\hat{VR}^{XSM}_{jt} (\Theta) \right)^2$$</span></p>
<p>Although, you should still take the estimates of such model with a little bit of a grain of salt as it does not necessarily solve all issues in such estimations, but it would give you at least reasonable estimates for how firm value depends on capital and labor.</p>

CC BY-SA 4.0
2021-07-24T11:24:34.280Z
2021-07-24T12:52:05.273Z
2021-07-24T12:52:05.273Z
1muflon1
<p>How do I determine how much value that capital, labor, and land individually contributed to the value of a firm/good? Like what % of value came from capital vs labor vs land? I heard that this is possible in neoclassical economics, but I'm not sure how. Like say I sold chairs. The market rate of wood is 10 units per wood for a chair. Laborers are paid 5 units. Chairs are sold for 20 units.</p>
<p>Did labor contribute 10 units of value? Or is the value of risk capital took plus investment in factories and machines for chair manufacturing factored in as well ?</p>
<p>Thanks!</p>

<producer-theory>
How do you calculate the contribution of each factor of production to the value of the final product?
Robotic_Cow
# How do you calculate the contribution of each factor of production to the value of the final product?

(Asked by: username_0 on 2021-07-24)

How do I determine how much value that capital, labor, and land individually contributed to the value of a firm/good? Like what % of value came from capital vs labor vs land? I heard that this is possible in neoclassical economics, but I'm not sure how. Like say I sold chairs. The market rate of wood is 10 units per wood for a chair. Laborers are paid 5 units. Chairs are sold for 20 units.

Did labor contribute 10 units of value? Or is the value of risk capital took plus investment in factories and machines for chair manufacturing factored in as well ?

Thanks!

**Tags:** producer-theory

---

## 2 answers

### Answer by username_4 on 2021-07-24. Score: 3 | Accepted answer

You can do this in various ways for a *value of output* (e.g. goods and services) firm produces but not really very reliably for value of firm itself.

**Value of Output**

When it comes to value of output then this boils down to estimating production function of the company. For example, assuming that company has Cobb-Douglas production function:

$$Y = AK^\beta L^{(1-\beta)}$$

where $Y$ is the value of output, $A$ is the multifactor productivity, $K$ is the capital, and $L$ labor input respectively. You can just log-linearize the expression (below I am using common notation for logs in economics $x= \ln X$) and turn it into

$$ y = a + \beta k + (1-\beta) l $$

this can be then set up as a panel regression model regression model (where $a$ is decomposed to $\beta\_0$, $\omega\_{it}$ and $\epsilon\_{it}$:

$$ y = \beta\_0 + \omega\_{it} + \beta\_1 k\_{it} + \beta\_2 l\_{it} + \epsilon\_{it}$$

However, you cannot just use naïve panel regression as capital is incredibly difficult to properly measure, so you will likely be only able to find accounting data which are created for a purpose of minimizing firm's tax obligations not to accurately describe capital stock of company or its input utilization.

Thankfully, over the years various methodologies and empirical strategies were developed to overcome this issue (although it is not 100% solved problem). The main solutions to this problem are the so called [Olley Pakes (1996)](https://www.jstor.org/stable/2171831?seq=1#page_scan_tab_contents), [Levinsohn-Petrin (2000)](https://www.nber.org/papers/w7819) and [Wooldridge (2009)](https://www.sciencedirect.com/science/article/abs/pii/S0165176509001487?via%3Dihub) models (all are variants on the general method of moments [GMM]). These models use various instruments to correct for the endogeneity issue caused by capital measurement errors (although again it's not 100% solved issue but the above methods are state of the art).

**Firm Value**

When it comes to value of firm, that is trickier. From an economic perspective value of firm depends almost solely on expectations of its future profitability and streams of cash it creates through dividends paid from the said profit (see discussion in Mishkin and Eakins Financial Markets and Institutions in the chapter on stock market). Firms like Nokia or Blockbuster or Kodak had both large capital stocks and number of employees even at a point where firm value was driven to near zero.

Even though profitability on some level depends use of capital and labor the relationship is much less clear cut when it comes to firm value as when it comes to firm output. In addition you have the same capital (mis)measurement problem as in the previous section, but now with even less good models to suggest appropriate instruments to correct for it.

However, there were recent attempts at doing this. For example, [Belo et al (2019)](https://www.nber.org/system/files/working_papers/w26112/w26112.pdf) have recent working paper on this topic. They estimate model based on following firm valuation ratio:

$$VR\_{it} = q^P\_{it}\frac{K^P\_{it+1}}{A\_{it+1}} + q^L\_{it}\frac{L\_{it+1}}{A\_{it+1}}+ q^K\_{it}\frac{K^K\_{it+1}}{A\_{it+1}} + q^B\_{it}\frac{K^B\_{it+1}}{A\_{it+1}} \tag{\*}$$

where $VR$ is the firm valuation ratio ($V R\_{it} \equiv (P\_{it} + B\_{it+1}) /A\_{it+1}$), where $P$ is the firm's stock price, $B$ is the firm level debt, and $A$ is the measure of firm effective assets. Next, $q^j, j=P,K,B,L$ are shadow prices for the firm price, two types of capital stocks (physical and brand capital $K^K, K^B$) and labor which can be inferred from firm level data, K are various measures of capital (depending on superscript) and $L$ is labor stock. They estimate the model parameters by minimizing the squared distance between the portfolio-level
observed and model-implied valuation ratio moments at each point in time.

That is they first set up the relationship between observed valuation ratios given by:

$$VR^{XSM}\_{jt} (\Theta) = \sum\_i \frac{\hat{V R}\_{it}}{N\_{jt}}, i \in \text{ portfolio } j$$,

where the left hand side has observed valuations (that depend on parameters $\Theta$ that are parameters of the model given by (\*), and right hand side valuations to be estimated, which are later estimated setting up:

$$ VR^{XSM}\_{jt}= \hat{VR}^{XSM}\_{jt} (\Theta) + \epsilon\_{it}$$

where $\epsilon$ captures measurement error in the portfolio moments, and the parameters are in the end estimated using:

$$\hat{Θ }= \arg \min\_{\Theta} \sum\_{t=1}^T \sum\_{n=1}^N \left( VR^{XSM}\_{jt}-\hat{VR}^{XSM}\_{jt} (\Theta) \right)^2$$

Although, you should still take the estimates of such model with a little bit of a grain of salt as it does not necessarily solve all issues in such estimations, but it would give you at least reasonable estimates for how firm value depends on capital and labor.


---

### Answer by username_1 on 2021-07-24. Score: -2

According to the discussion in the seven page article available under this link (splash page):

<https://nsereview.org/index.php/NSER/article/view/79>

> Using a Cobb-Douglas production function, regressions found a fairly good fit to the data and appeared to confirm the marginal productivity theory of distribution. While some were puzzled by the seemingly too-good-to-be-true empirical confirmation of a theory with shaky foundations, Anwar Shaikh demonstrated forty years ago that “so long as aggregate shares are constant, an aggregate Cobb-Douglas function having apparently ‘constant returns to scale’ will always provide an exact fit, for any data whatsoever” and “will seem to also possess ‘marginal products equal to respective factor rewards’”. Any data! From any kind of economy, including a Humbug economy! The results are guaranteed not due to laws of economics but rather due to laws of algebra.

> Robert Solow replied to Anwar as follows: The factor-share device of my 1957 article is in no sense a test of aggregate production functions or marginal productivity or anything else. It merely shows how one goes about interpreting given time series if one starts by assuming that they were generated from a production function and that the competitive marginal-product relations apply. [Solow 1974: 121, emphasis in original - Solow, R.M.1974, “Law of production and laws of algebra: the Humbug production function: A comment”, Review of Economics and Statistics, 56(1).]

The aggregate production function models are not considered to be empirical measures. These models make it appear as though aggregate income is or could be a function of the marginal contribution of labor and capital. The article states that such models can fit a curve to any data which could have been generated by an aggregate production function with the specified parameters. The post-keynesian argument is that there is no singular production process, there is no empirical measure of aggregate capital, and there is no deterministic model for the historical allocation of aggregate income to shares of labor and profit. Instead there is a complex interaction among economic and political agents, an outcome, and curve fitting exercise called the production function.

> **Comments:**
> **username_2**: You seem to be commenting on the [other answer](https://economics.stackexchange.com/a/45954/1601), not trying to answer the question itself.  
> **username_1**: The question includes this comment "I heard that this is possible in neoclassical economics, but I'm not sure how." The "how" in general is to assume some production function Y = f(K,L) where income is a function of the contribution of capital and the contribution of labor. The meaning of these functions is the subject of debate in the field of economics. My answer relates to the meaning of the assumptions applied in those methods which is more important to understand in terms of economic reasoning than the exact algebraic method.  
> **username_2**: I see; alright.  
> **username_3**: this does not answer the question, the source and quote talks about aggregate production function, but the question ask about micro level individual production function  
> **username_1**: Neither micro nor macro production functions are empirical measurements because there is no objective way to measure of the contribution of capital compared to the contribution of labor. Production functions should be recognized as curve fitting models and with no deterministic model for describing the empirical (measured) contribution of capital and the empirical (measured) contribution of labor to the output produced.  

---

