Cobb–Douglas production function ($f(K,L)=A{K}^{\alpha }{L}^{\beta }$)

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 $(\alpha +\beta )\varphi (b)/b$, 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.