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Theoretical Economics Paper
The Wong-Viner Envelope Theorem for subdifferentiable functions
Anthony Horsley and Andrew J Wrobel
April 2005
Paper No' TE/2005/489:
Full Paper (pdf)

JEL Classification: C61; D21;D41

Tags: wong-viner envelope theorem; nondifferentiable joint costs; profit-imputed valuation of fixed inputs; general equilibrium; public utility pricing.

The Wong-Viner Envelope Theorem on the equality of long-run and short-run marginal costs (LRMC and SRMC) is reformulated for convex but generally nondifferentiable cost functions. The marginal cost can be formalized as the multi-valued subdifferential a.k.a. the subgradient set but, in itself, this is insufficient to extend the result effectively, i.e., to identify suitable SRMCs as LRMCs. This goal is achieved by equating the profit-imputed values of the fixed inputs to their prices. Thus reformulated, the theorem is proved from a lemma on the sections of the joint subdifferential of a bivariate convex function. The new technique is linked to the Partial Inversion Rule of convex calculus.