Why in Cobb-Douglas utility function, the exponents have to sum to one? Can they not be equal to 1 and why?
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1$\begingroup$ They can. Why? Because it makes sense mathematically and economically. Why it makes sense depends on what the CD functions are used to model. Standardly, economists use the CD function to model either preferences or production functions. For modeling preferences positive monotone transformations of the CD function are allowed, hence it whether coefficients sum to one or not is without consequence. This is not the case with production functions the sum of the coefficients determines whether the production technology displays, decreasing, constant, or increasing returns to scale. $\endgroup$– Jesper HybelCommented Feb 17 at 19:51
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They don't have to, look: $U(x,y) = x^2y^2$ is Cobb-Douglas type, but the exponents sum up to four. It is also true that $U()$ represents the same preferences as $\hat{U}(x,y) = \sqrt{xy}$, where the exponents do sum up to one, and it is always possible to monotonically transform a Cobb-Douglas utility function in such a way that the exponents will sum up to one.
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$\begingroup$ every Cobb-douglas functions which have the sum of exponents higher than 1, can all be rewritten as a sum of 1, right? $\endgroup$ Commented Feb 18 at 3:59
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$\begingroup$ Rewritten is not precise enough, I specifically wrote monotonic transformation. You can look it up if you have not learned about it yet. $\endgroup$– GiskardCommented Feb 18 at 5:43