My Math Forum Shortcuts to Determining Returns to Scale

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 February 10th, 2012, 01:02 PM #1 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 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?
 February 11th, 2012, 06:03 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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.
 February 12th, 2012, 12:26 AM #3 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 Re: Shortcuts to Determining Returns to Scale What if there are two variables?
 February 12th, 2012, 04:43 AM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Shortcuts to Determining Returns to Scale You'll have to use partial derivatives.
 February 12th, 2012, 04:59 AM #5 Member   Joined: Nov 2009 Posts: 90 Thanks: 0 Re: Shortcuts to Determining Returns to Scale I see. I think I'll manage. Thanks for the clear answer!

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