Non-constant returns to scale and competitive factor markets I am having difficulties to understand the conclusion in bold below, taken from Frankel (1962):


  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 entry.

One interesting answer to this might come from the first answer here. 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?
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 $k&gt;1$ 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.

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.