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The Question

Suppose a firm has a production function given by

$$y=F(L,K)=L^{1/4}K^{1/4}$$

where L and K denote inputs used in the production of y units of output.

(a) Determine whether marginal products are diminishing

(b) Show that production technology exhibits decreasing returns to scale

My attempt

(a) So the marginal products, $MP_L$ , $MP_K$ are:

$$MP_L={\partial{F}\over\partial{L}}={1\over{4}}L^{-3\over{4}}K^{1\over{4}}$$

$$MP_k={\partial{F}\over\partial{K}}={1\over{4}}L^{1\over{4}}K^{-3\over{4}}$$

To determine if the marginal products are diminishing one needs to simply derive the equations again. Which would be:

$${\partial{MP_L}\over{\partial{L}}}={-3\over{16}}L^{-7\over{4}}K^{1/4}$$

and

$${\partial{MP_k}\over{\partial{K}}}={-3\over{16}}L^{1\over{4}}K^{-7\over{4}}$$

When both Marginal products are derived, their results are both are $<0$ which would imply that they are diminishing.

(b) This is where I get a little confused, is it not because we know that the Marginal products are diminishing, we know that the production technology exhibits decreasing returns to scale?

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  • $\begingroup$ Marginal product is about how output changes when only one input changes. Returns to scale is about how output changes when all inputs change in same proportion. $\endgroup$ – Paul Oct 30 '16 at 20:29
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They are two different concepts I think.

Diminishing product means that holding other things fixed, one unit of extra input($K$ or $L$ here) yields less and less additional output, which you have known.

Meanwhile DRS is saying that if you multiply both $K$ and $L$ by some scalar $ t > 1$, the corresponding output would be less than $t$ times the original output.

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  • $\begingroup$ so then I simply assign any +ive value $\forall{L,K}$ and it would demonstrate such? $\endgroup$ – FreakconFrank Oct 30 '16 at 20:33
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    $\begingroup$ @PloniAlmoni $\forall t>1$, $F(tK, tL) = (tK)^{1/4}(tL)^{1/4} = t^{1/4+1/4}F(K, L)$. So you can see it does exhibit DRS here. In fact, in this form of production function you can easily see that $L^\alpha K^\beta$ will be DRS whenever $\alpha+\beta<1$. $\endgroup$ – Eric Chen Oct 30 '16 at 20:40

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