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February 10th, 2012, 01:02 PM   #1
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Shortcuts to Determining Returns to Scale

When determining if a production function (or whatever) has decreasing, constant or increasing returns to scale, all of my sources advise me to multiply each factor by a constant greater than 1 and see how that affects the production. Sometimes however, the functions are quite complex and it might be worthwhile learning other systematic general methods, perhaps based on calculus. So my question is whether such a method exists?

A Cobb-Douglas Production function y=A(x^a)(y^b) has decreasing RTS if a+b<1, constant RTS if a+b=1 and increasing RTS if a+b>1. But what if the function is not in the Cobb-Douglas form? For example, let's say I have y=(ax+bx)^c where a, b and c are constants.

I figured out that if all factors are strictly convex, returns to scale must be increasing. So I'm thinking the second partial derivatives of all factors can say something about RTS?
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February 11th, 2012, 06:03 PM   #2
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Re: Shortcuts to Determining Returns to Scale

If the second partial is positive over the entire range of interest, the returns are increasing to scale.
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February 12th, 2012, 12:26 AM   #3
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Re: Shortcuts to Determining Returns to Scale

What if there are two variables?
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February 12th, 2012, 04:43 AM   #4
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Re: Shortcuts to Determining Returns to Scale

You'll have to use partial derivatives.
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February 12th, 2012, 04:59 AM   #5
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Re: Shortcuts to Determining Returns to Scale

I see. I think I'll manage. Thanks for the clear answer!
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