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This question already has an answer here:

I know that diminishing marginal returns even to all factors of production doesn't imply decreasing returns to scale. But could you please give me just an example of such production function?

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marked as duplicate by luchonacho, Giskard, Alecos Papadopoulos, Herr K., emeryville Sep 25 '17 at 20:27

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  • $\begingroup$ In the title you write "increasing", in the question you write "decreasing". Correct and clarify. $\endgroup$ – Alecos Papadopoulos Sep 24 '17 at 23:30
  • $\begingroup$ This is not a duplicate. In the linked to problem, marginal productivity is not (strictly) decreasing. $\endgroup$ – Michael Greinecker Sep 26 '17 at 10:34
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Decreasing marginal returns to a factor means that keeping the other factors fixed, the marginal output generated by this factor is decreasing. When looking at returns to scale, we change all outputs. Increasing a factor with decreasing marginal returns can have an indirect effect in increasing the marginal productivity of other factors. If we increase all factors at the same time, the indirect effects may outweigh the direct effect. The production function $F:\mathbb{R}_+^2$ given by $$F(x,y)=(x+1)^{2/3}(y+1)^{2/3}$$ has decreasing marginal factor productivities everywhere but not decreasing returns to scale (it doesn't have increasing returns to scale either).

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For decreasing marginal returns we require second partial derivatives to be negative, since we examine what happens if we vary only one input

So any function $$y = \prod_{i=1}^{m}x_i^{a_i},\;\;\; 0<a_i<1\;\; \forall \,i$$

with $\sum a_i >1$ for increasing returns to scale, and with $\sum a_i <1$ for decreasing returns to scale, since here we examine what happens of we increase by the same proportion all inputs.


Responding to comments

We check returns to scale for this function by examining, for $k>1$, the expression

$$ \prod_{i=1}^{m}(kx_i)^{a_i} = k^{\sum a_i}\cdot y$$

If the sum of the alphas is higher than unity, output increases more than $k$ so we have increasing returns to scale, and correspondingly for deceasing returns to scale when the sum of the alphas is smaller than unity.

Regarding decreasing marginal returns, the rate of change of marginal output generated by a factor keeping the others fixed is given by its own second partial derivative, which here is

$$\frac {\partial^2 y}{\partial x_i^2}= a_i(a_i-1)\cdot \frac {y}{x_i^2}$$

When $0<a_i<1$, these second partials are all negative.

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  • $\begingroup$ What is your definition of decreasing/increasing returns to scale? For example, under the definition used in the Wikipedia entry on returns to scale, a Cobb-Douglas production function cannot have positive or decreasing returns to scale (something the entry later gets wrong). $\endgroup$ – Michael Greinecker Sep 25 '17 at 5:47
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    $\begingroup$ @MichaelGreinecker I expanded my answer to answer your questions. $\endgroup$ – Alecos Papadopoulos Sep 25 '17 at 6:46
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    $\begingroup$ @MichaelGreinecker I certainly do, since it is a mathematical case which is rather uninteresting from the point of view of Economics. As regards the definition in wikipedia for the Cobb-Douglas function, it follows the convention to call "Cobb-Douglas" the function when the sum of exponents equals exactly unity, while when we want to use the exponential form with sum of exponents different than unity we usually call it "generalized Cobb-Douglas". $\endgroup$ – Alecos Papadopoulos Sep 25 '17 at 6:57
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    $\begingroup$ @MichaelGreinecker I am not sure that I understand what more should I write, except perhaps by adding the qualifier $x_i>0,\, \forall\, i$. All Economics textbooks I know don't even bother to mention that, exactly because a "zero-output case due to fixing an input to zero" is uninteresting for the purposes of Economics. $\endgroup$ – Alecos Papadopoulos Sep 25 '17 at 7:11
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    $\begingroup$ I do hold the view that all economic textbooks can be wrong, in particular, those that make claims to mathematical rigor. A textbook that does not commit the mistake is "Microeconomic Foundations I" by David Kreps. $\endgroup$ – Michael Greinecker Sep 25 '17 at 7:40

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