tailieunhanh - Real Analysis with Economic Applications - Chapter E

Chapter E Continuity II A function that maps every element of a given set to a nonempty subset of another set is called a correspondence (or a multifunction). Such maps arise quite frequently in optimization theory and theoretical economics. While this is not really a standard topic in real analysis, this chapter is devoted to the analysis of correspondences | Chapter E Continuity II A function that maps every element of a given set to a nonempty subset of another set is called a correspondence or a multifunction . Such maps arise quite frequently in optimization theory and theoretical economics. While this is not really a standard topic in real analysis this chapter is devoted to the analysis of correspondences because of their importance for economists. Our first task is to understand in what sense a correspondence can be viewed as continuous. After a brief set-theoretical overview of correspondences we thus spend some time examining various continuity concepts for correspondences. These concepts are needed to state and prove Berge s Maximum Theorem which tells one when the solution to an optimization problem depends on the parameters of the problem in a continuous way. Along with a few relatively straightforward applications that attest to the importance of this result the chapter touches also upon a major topic of optimization theory namely the theory of stationary dynamic programming. We discuss in particular the issue of existence uniqueness and monotonicity of solutions of a dynamic programming problem at some length and illustrate our findings through a standard topic in macroeconomics namely the one-sector optimal growth model. Delving into the theory of continuous correspondences more deeply the rest of the chapter is a bit more advanced than the earlier sections. In this part we introduce the partition of unity and prove the famous Michael Selection Theorem which provides sufficient conditions to find a continuous function within a given correspondence in the sense that the graph of the function is contained in that of the correspondence . In turn by using this result and the Brouwer Fixed Point Theorem we derive the celebrated Kakutani Fixed Point Theorem which is used frequently in equilibrium analysis. While it is of a different flavor Nadler s Contraction Correspondence Theorem also gets some attention in