Talk:Cobb-Douglas

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I cannot understand why it is that when α + β = 1 there are constant returns to scale. Does anybody know how to derive this?

Sure. First of all, if α + β = 1, then β = 1 - α, so we'll substitute that in. Also, let's say that n is the factor that we're scaling by. So our initial setup is

A({\color{Blue}n}L)^\alpha\,({\color{Blue}n}K)^{1-\alpha\,}

A{\color{Blue}n}^\alpha\,L^\alpha\,{\color{Blue}n}^{1-\alpha\,}K^{1-\alpha\,} // distribution of exponents

{\color{Blue}n}^\alpha\,{\color{Blue}n}^{1-\alpha\,}AL^\alpha\,K^{1-\alpha\,} // just reordering things

{\color{Blue}n}^{\alpha\,+1-\alpha\,}AL^\alpha\,K^{1-\alpha\,} // multiplication with exponents

{\color{Blue}n}^1AL^\alpha\,K^{1-\alpha\,}

{\color{Blue}n}AL^\alpha\,K^{1-\alpha\,}

so the result of scaling up both inputs by n is that the final answer is also scaled up by n. --152.23.101.108 02:30, 2 September 2006 (UTC)

(Or, more abstractly, scaling up both inputs by n results in the output being scaled up by n^{\alpha\,+\beta\,})

The above might be too specific to put on the page, but maybe we could say something about how the function exhibits homogeneity of degree 1 when the exponents sum to 1? That way we can link to the mathematics article that explains the concept a bit more succinctly (thus avoiding those sort of derivations when unnecessary). Bsdlite (talk) 21:31, 14 May 2008 (UTC)


This seems a bit unclear: "A, α and β are the output elasticities of labor and capital, respectively. These values are constants determined by available technology. " —Preceding unsigned comment added by 61.230.54.247 (talk) 19:30, 18 May 2008 (UTC)