General equilibrium with set-valued supply and demand

Our plan in this chapter is to take the model of production, consumption, the economy, and market equilibrium of Chapters 15–18 and restate it for the case of set-valued demand and supply behavior. Formally this means that we dispense with assumptions of strict convexity of tastes and production technology, C.VI(SC) and P.V. We rely rather on convexity, C.VI(C) and P.I. Under the remaining assumptions on consumption and production behavior, this will allow us to characterize demand and supply behavior as upper hemicontinuous, convex-valued correspondences. In turn, excess demand will then be characterized as upper hemicontinuous and convex-valued. A model of price adjustment that is also upper hemicontinuous and convex valued completes the picture: Applying the Kakutani Fixed-Point Theorem allows us to find a fixed point in price space that achieves a market equilibrium.

Just as we did in Chapters 15–18, we treat the economy in two formats: an artificially restricted bounded economy denoted by the superscript tilde notation (~) and an unrestricted economy (representing the true model we are really interested in). The artificially restricted economy is a purely technical construct, designed to allow us to develop the properties of the underlying unrestricted economy in a more tractable setting. The technique of the proof is to note that the restricted budget, demand, supply, and profit behavior is always well defined because it represents optimizing behavior on a compact set.