# Is “$f(k,l) \: is \: decreasing\:return\:to\:scale \Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$” always true?

For the productions $$f(k,l)$$ that are continuously differentiable, is the proposition that "$$f(k,l) \: is \: decreasing\:return\:to\:scale \Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$$" always true, I have checked for Cobb-Douglas functions and believe it will also hold for all Constant-Elastisity fuction, can someone give me some hints or show some counter-examples?

Here we define $$f(k,l)$$ to be decreasing return to scale if $$f(tk,tl)1\: for\: \forall (k,l)\:in \:the\:domain$$

For a single output production function, decreasing returns to scale does imply concavity of $$f$$. I believe you can find this result in Ch5 of MWG, either as a proposition or an exercise. The condition you have comes from considering the Hessian of $$f$$, and showing that the determinant is positive. For more on the Hessian Conditions, this is a handy reference: https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/cvn/t