= Cobb–Douglas production function
{c}
{title2=$f(K,L)=AK^\alpha L^\beta$}
{wiki}
For positive constants $A,\alpha,\beta$, a Cobb–Douglas production function describes output in terms of capital $K$ and labour $L$. It is homogeneous of degree $\alpha+\beta$. Under $K+wL=b$ with $b,w>0$, its unique nonnegative-input optimum is $K=\alpha b/(\alpha+\beta)$ and $L=\beta b/[w(\alpha+\beta)]$, obtained by maximizing the strictly concave logarithm along the budget line. The optimized output has derivative $(\alpha+\beta)\phi(b)/b$, equal to its budget <Lagrange multiplier>. When $\alpha+\beta>1$, the unconstrained Lagrangian supremum over the positive orthant is infinite for every finite multiplier, despite the finite constrained optimum; the global dual representation must not be presumed in this case.
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