Differentiable Technology, the Curvature of the Profit Function, and the Response of Supply to Own-Price Changes

Abstract

The author establishes a property of supply for a competitive firm: Assuming differentiability of the production frontier, linearly independent price vectors have disjoint image sets under the supply mapping. This property supports the main results. First, the author drew a simple proof of McFadden's proposition that differentiability of the production frontier is necessary and sufficient for strict quasiconvexity of the profit function. This proposition rules out linearity of the profit function on any subset of price domain, in turn ruling out supply and input- demand functions that are constant and have zero values for all price effects. Second, for discrete price changes, own-price effects in supply are strictly positive, assuming a differentiable technology. In this context, strict convexity of the production set is irrelevant. The author indicates implications for cost and demand theory.

Keywords:

• convexity
• cost function
• differentiability
• expenditure function
• Hotelling's lemma
• own-price effect
• profit function
• strict quasi convexity JEL codes: D21
• D11