After studying the behavior of optimizing agents, in this lecture we turn to the analysis of economies such agents form. As we shall see, there is a sharp demarcation between systems composed of perfect and distorted markets. With no distortions, we can take the social planner approach to be studied in Chapter 16. In the latter case, we need other techniques (to be studied in Chapters 17-18). The first line is based on the two welfare theorems: A competitive equilibrium results in a Pareto-efficient allocation, and any Pareto-efficient allocation can be underpinned as a competitive equilibrium by an equilibrium price system. We need to learn more about the conditions under which Pareto-efficient allocations and competitive equilibria coincide exactly. Most importantly, we shall see that to turn an efficient allocation into a competitive equilibrium we need an appropriate price system. When we solve a dynamic problem, we seek the solution as a point in an appropriately chosen commodity space. Price systems live in the dual space of the commodity space, so price systems are taken to be linear functionals––they are supposed to be interpretable as series of prices. A commodity space is a normed vector space, and it establishes the properties of its dual. There is a fly in the ointment, however. Even if there are duals corresponding to specific commodity spaces in which the linear functionals always have inner product representations necessary to be interpretable as price systems, there might be linear functionals on a normed vector space that don’t lie in the dual. At the end of Part A of Lecture 14 we reach the Hahn–Banach theorem that establishes the conditions necessary for a price system.

Title page:(00:00)
Introduction:(00:10)
Dual spaces:(07:07)
Exercise 15.3:(17:05)
Exercise 15.4:(27:37)
The Hahn-Banach theorem:(41:16)
Exercise 15.6:(47:08)
Problems in the choice of a commodity space:(55:46)