<p>Frankel examines the use of the Cobb-Douglas formulation in <strong>aggregate</strong> data, <em>where as a matter of macroeconomic(and essentially logical) identity output is exhausted in paying the factors of production</em>.</p>

<p>So the macroeconomic identity states, for <em>any</em> aggregate production function</p>

<p>$$Q=F(K,L) \equiv rK + wL$$</p>

<p>where $r,w$ are <em>ex post</em> average unit rewards for the factors of production (hence the identity character).</p>

<p>The <em>only</em> way to equate $r$ and $w$ with marginal product is to have a function homogeneous of degree one ("constant returns to scale") because it is only then that we have</p>

<p>$$F(K,L) = F_K\cdot K + F_L\cdot L$$</p>

<p>and we can map $r=F_K,\; w=F_L$.</p>

<p>If the function is homogeneous but not of degree one but of a degree $k\neq 1$, then we have </p>

<p>$$F(K,L) = \frac 1k(F_K\cdot K + F_L\cdot L)$$</p>

<p>which leads to $r=(1/k)F_K,\;\; w=(1/k)F_L$</p>

<p>If $k&gt;1$ we have Increasing returns to scale and factors are paid less than their marginal product (again, on average and on aggregate), while if $k&lt;1$ we have Decreasing returns to scale and factors are paid more than their marginal product.</p>

<p>We then run into trouble regarding the <em>behavior</em> and <em>structure</em> assumptions at the micro level that lead to such a result at the macro-level.</p>

CC BY-SA 3.0
2017-08-18T15:25:26.400Z
After actually reading the full paper, the author states (page 1002) that "Similarly, the use of an 
aggregate 
function 
with 
nonconstant 
returns 
to 
scale 
need 
not 
carry 
any 
implication 
that 
factors 
don't 
get 
their 
marginal 
products". That is because he assumes that technical level at the firm level ($H$) is a function of aggregate capital per worker, which leads to an aggregate production function with increasing returns to scale, while keeping constant returns to scale at the firm level. This is of course a very used trick in the growth literature.
luchonacho
CC BY-SA 3.0
2017-08-18T15:27:53.400Z
Strictly speaking, we do not need to abandon competitive assumptions under IRS. You can update (correct?) your answer with that insight, or I can provide another, if you prefer.
luchonacho
CC BY-SA 3.0
2017-08-18T15:47:16.420Z
@luchonacho Allowing for externalities is a special case, and I believe it would be better to write a separate answer to present this case, leaving mine to deal with the baseline case.
Alecos Papadopoulos
CC BY-SA 3.0
2017-06-16T16:04:29.960Z
@luchonacho you ask microeconomic questions while the issue Frankel discusses is at the macroeconomic level. Frankel asks: if the aggregate production function appear empirically to have increasing or decreasing returns to scale, what does this imply for what happens at the micro- and firm level?
Alecos Papadopoulos
CC BY-SA 3.0
2017-06-16T16:07:16.897Z
@luchonacho Reality does not have to explain itself to us.
Alecos Papadopoulos
CC BY-SA 3.0
2017-06-15T18:35:25.613Z
2017-06-15T20:18:54.780Z
2017-06-15T20:18:54.780Z
Alecos Papadopoulos
<p>I am having difficulties to understand the conclusion in bold below, taken from <a href="https://eclass.uoa.gr/modules/document/file.php/ECON206/Course%20material/Frankel%20Production%20function%20AER%201962.pdf" rel="nofollow noreferrer">Frankel (1962)</a>:</p>

<blockquote>
  <p>A  second  limitation  of  the  Cobb-Douglas  function  appears  when  it  is  fitted  to  historical  data.  All  improved  fit  generally  is  obtainable  if  the  exponents:  and  a  are  allowed  to  vary  freely  instead  of  being  constrained  to  equal  unity.  With  some  sets  of  data,  the  resulting  sum  of  the  exponents  has  differed  significantly  from  unity,  an  outcome  that  opens  the  door  to  economies  and  diseconomies  of  scale  and  that  leads  to  <strong>abandonment  of  the  convenient  assumption  that  factors  are  paid  their  marginal  products.</strong> </p>
</blockquote>

<p>As I understand things, returns to scale is a technical issue, which <em>might</em> be independent on market conditions. Albeit not a reliable source, the same appears in the <a href="https://en.wikipedia.org/wiki/Returns_to_scale" rel="nofollow noreferrer">Wikipedia</a> entry.</p>

<p>One interesting answer to this might come from the first answer <a href="https://www.quora.com/Why-is-increasing-return-to-scale-incompatible-with-perfect-competition" rel="nofollow noreferrer">here</a>. However, the author is interested in competition among producers, in terms of the final goods price. Basically, that a natural monopoly (where IRS hold), has an average cost always below the marginal cost, so competitive market produces loses. <strong>But what prevents a natural monopoly to pay its factors their marginal products?</strong> You can think of a natural monopoly, or any other example of firms with DRS or IRS, in which they operate amid competitive capital and labour markets. I see no <em>a priori</em> reason why market forces would not lead these firms to pay the factors their marginal product. </p>

<p>To sum up: why IRS or DRS mean we should abandon competitive assumptions <strong>in the factor markets</strong>?</p>

CC BY-SA 3.0
2017-08-18T15:30:43.550Z
@denesp see new comments on Alecos' answer.
luchonacho
<microeconomics><competition>
Non-constant returns to scale and competitive factor markets
luchonacho
# Non-constant returns to scale and competitive factor markets

(Asked by: username_0 on 2017-06-15)

I am having difficulties to understand the conclusion in bold below, taken from [Frankel (1962)](https://eclass.uoa.gr/modules/document/file.php/ECON206/Course%20material/Frankel%20Production%20function%20AER%201962.pdf):

> A second limitation of the Cobb-Douglas function appears when it is fitted to historical data. All improved fit generally is obtainable if the exponents: and a are allowed to vary freely instead of being constrained to equal unity. With some sets of data, the resulting sum of the exponents has differed significantly from unity, an outcome that opens the door to economies and diseconomies of scale and that leads to **abandonment of the convenient assumption that factors are paid their marginal products.**

As I understand things, returns to scale is a technical issue, which *might* be independent on market conditions. Albeit not a reliable source, the same appears in the [Wikipedia](https://en.wikipedia.org/wiki/Returns_to_scale) entry.

One interesting answer to this might come from the first answer [here](https://www.quora.com/Why-is-increasing-return-to-scale-incompatible-with-perfect-competition). However, the author is interested in competition among producers, in terms of the final goods price. Basically, that a natural monopoly (where IRS hold), has an average cost always below the marginal cost, so competitive market produces loses. **But what prevents a natural monopoly to pay its factors their marginal products?** You can think of a natural monopoly, or any other example of firms with DRS or IRS, in which they operate amid competitive capital and labour markets. I see no *a priori* reason why market forces would not lead these firms to pay the factors their marginal product.

To sum up: why IRS or DRS mean we should abandon competitive assumptions **in the factor markets**?

**Tags:** microeconomics, competition
> **Comments:**
> **username_0**: @denesp see new comments on Alecos' answer.  

---

## 1 answers

### Answer by username_1 on 2017-06-15. Score: 4 | Accepted answer

Frankel examines the use of the Cobb-Douglas formulation in **aggregate** data, *where as a matter of macroeconomic(and essentially logical) identity output is exhausted in paying the factors of production*.

So the macroeconomic identity states, for *any* aggregate production function

$$Q=F(K,L) \equiv rK + wL$$

where $r,w$ are *ex post* average unit rewards for the factors of production (hence the identity character).

The *only* way to equate $r$ and $w$ with marginal product is to have a function homogeneous of degree one ("constant returns to scale") because it is only then that we have

$$F(K,L) = F\_K\cdot K + F\_L\cdot L$$

and we can map $r=F\_K,\; w=F\_L$.

If the function is homogeneous but not of degree one but of a degree $k\neq 1$, then we have

$$F(K,L) = \frac 1k(F\_K\cdot K + F\_L\cdot L)$$

which leads to $r=(1/k)F\_K,\;\; w=(1/k)F\_L$

If $k>1$ we have Increasing returns to scale and factors are paid less than their marginal product (again, on average and on aggregate), while if $k<1$ we have Decreasing returns to scale and factors are paid more than their marginal product.

We then run into trouble regarding the *behavior* and *structure* assumptions at the micro level that lead to such a result at the macro-level.

> **Comments:**
> **username_1**: @luchonacho you ask microeconomic questions while the issue Frankel discusses is at the macroeconomic level. Frankel asks: if the aggregate production function appear empirically to have increasing or decreasing returns to scale, what does this imply for what happens at the micro- and firm level?  
> **username_1**: @luchonacho Reality does not have to explain itself to us.  
> **username_0**: After actually reading the full paper, the author states (page 1002) that "Similarly, the use of an  aggregate  function  with  nonconstant  returns  to  scale  need  not  carry  any  implication  that  factors  don't  get  their  marginal  products". That is because he assumes that technical level at the firm level ($H$) is a function of aggregate capital per worker, which leads to an aggregate production function with increasing returns to scale, while keeping constant returns to scale at the firm level. This is of course a very used trick in the growth literature.  
> **username_0**: Strictly speaking, we do not need to abandon competitive assumptions under IRS. You can update (correct?) your answer with that insight, or I can provide another, if you prefer.  
> **username_1**: @luchonacho Allowing for externalities is a special case, and I believe it would be better to write a separate answer to present this case, leaving mine to deal with the baseline case.  

---

