References and Literature
This chapter covered a lot of ground and in most cases, many details were omitted for brevity. Many readers will be familiar with some of the material in this chapter. Mas-Colell, Winston and Green (1995) provide an excellent discussion of issues of aggregation.
They present results similar to Theorems 5.2 and 5.3 above. The first converse to Theorem 5.2 was stated and proved for additive utility functions by Pollak (1971), but as the discussion following this theorem indicates, the same result applies without the need to assume additivity. Mas-Colell, Winston and Green also present a version of the Debreu-Mantel-Sonnenschein theorem, with a sketch proof and an excellent discussion.Some basic concepts from microeconomic theory were assumed in the discussion, and the reader can also find a thorough exposition of these in Mas-Colell, Winston and Green (1995). These include Roy’s identity, used in Theorem 5.3, and Walras’s Law, the concept of numeraire and expected utility theory of von Neumann and Morgenstern, used throughout the analysis.
The Representative Firm Theorem, Theorem 5.4, presented here is quite straightforward, but I am not aware of any other mention or discussion of this theorem in the literature. It is important to distinguish the subject matter of this theorem from the Cambridge controversy in early growth theory, which revolved around the issue of whether different types of capital goods could be aggregated into a single capital index (see, for example, Wan, 1969). The Representative Firm Theorem says nothing about this issue.
The best reference for existence of competitive equilibrium and the welfare theorems with a finite number of consumers and a finite number of commodities is still Debreu’s (1959) Theory of Value. This short book introduces all of the mathematical tools necessary for general equilibrium theory and gives a very clean exposition.
Equally lucid and more modern are the treatments of the same topics in Mas-Colell, Winston and Green (1995) and Bewley (2006). The reader may also wish to consult Mas-Colell, Winston and Green (1995, Chapter 16) for a proof of the Second Welfare Theorem with a finite number of commodities (Theorem 5.7 above is more general because it covers the case with infinite number of commodities). Both of these books also have an excellent discussion of the necessary restrictions on preferences so that they can be represented by utility functions. Mas-Colell, Winston and Green (1995, Chapter 19) also gives a very clear discussion of the role of Arrow securities and the relationship between trading at a single point in time and sequential trading. The classic reference on Arrow securities is Arrow (1964).Neither of these two references discuss infinite-dimensional economies. The seminal reference for infinite-dimensional welfare theorems is Debreu (1954). Stokey, Lucas and Prescott (1989, Chapter 15) presents existence and welfare theorems for economies with a finite number of consumers and countably infinite number of commodities. The mathematical prerequisites for their treatment are greater than what has been assumed here, but their treatment is both thorough and straightforward to follow once the reader makes the investment in the necessary mathematical techniques. The most accessible references for the Hahn-Banach Theorem, which is necessary for a proof of Theorem 5.7 in infinite-dimensional spaces, are Kolmogorov and Fomin (1970), Kreyszig (1978) and Luenberger (1969). The latter is also an excellent source for all the mathematical techniques used in Stokey, Lucas and Prescott (1989) and also contains much material useful for appreciating continuous-time optimization. Finally, a version of Theorem 5.6 is presented in Bewley (2006), which contains an excellent discussion of overlapping generations models.
5.12.