# 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?

• @denesp see new comments on Alecos' answer. – luchonacho Aug 18 '17 at 15:30

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.

• @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 Jun 16 '17 at 16:04
• @luchonacho Reality does not have to explain itself to us. – Alecos Papadopoulos Jun 16 '17 at 16:07
• 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 Aug 18 '17 at 15:25
• 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 Aug 18 '17 at 15:27
• @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 Aug 18 '17 at 15:47