TAILIEUCHUNG - Real Analysis with Economic Applications - Chapter H

Even the limited extent of convex analysis we covered in Chapter G endows one with surprisingly powerful methods. Unfortunately, in practice, it is not always easy to recognize the situations in which these methods are applicable. | Chapter H Economic Applications Even the limited extent of convex analysis we covered in Chapter G endows one with surprisingly powerful methods. Unfortunately in practice it is not always easy to recognize the situations in which these methods are applicable. To get a feeling for in which sort of economic models convex analysis may turn out to provide the right mode of attack one surely needs a lot of practice. Our objective in this chapter is thus to present a smorgasbord of economic applications that illustrate the multifarious use of convex analysis in general and the basic separation-by-hyperplane and linearextension arguments in particular. In our first application we revisit expected utility theory but this time using preferences that are potentially incomplete. Our objective is to extend both the classical and the Anscombe-Aumann Expected Utility Theorems Section into the realm of incomplete preferences and introduce the recently popular multi-prior decision making models. We then turn to welfare economics. In particular we prove the Second Welfare Theorem obtain a useful characterization of Pareto optima in pure distribution problems and talk about Harsanyi s Utilitarianism Theorem. As an application to information theory we provide a simple proof of the celebrated Blackwell s Theorem on comparing the value of information services and as an application to financial economics we provide various formulations of the No-Arbitrage Theorem. Finally in the context of cooperative game theory we characterize the Nash bargaining solution and examine some basic applications to coalitional games without side payments. While the contents of these applications are fairly diverse and hence they can be read independently of each other the methods with which they are studied here all stem from basic convex analysis. 1 Applications to Expected Utility Theory This section continues the investigation of expected utility theory we started in Section . We adopt here the

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