How accurate is duality?

Question

In economic theory we know that with the use of some calculus, Hotellings Lemma and Sheppards Lemma we can derive a given firms supply function and in term its Profit function.With data of a given firms costs, can we in fact get an accurate estimate of its profit function and supply function?

Answer 1

To determine the profit function we need to know the demand curve the the frim faces. If the firm is a price taker then it is straightorward.