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Introduction to Modern Economic Growth 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 full proof of the Second Welfare Theorem with a finite number of commodities (which was only sketched in Theorem 5.7 above) 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) also has an excellent discussion of expected utility theory of von Neumann and Morgenstern, which we have touched upon 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 the 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 reference 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 249