Example production function with increasing returns to scale but diminishing marginal product [duplicate]

Question

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?

Answer 1

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).

Answer 2

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.

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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.