Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.     

Gradients,subgradients, and economic equilibria

Two concepts of differentiability are shown to underlie two concepts of equilibrium for nonatomic economies. The weaker concept of subdifferentiability corresponds to Walrasian equilibrium and the stronger concept of Fréchet differentiability to product-exhaustion. Product-exhaustion is shown to be equivalent to a definition of economic equilibrium in terms of the rationality of price-taking.