2011 上财 高级微观经济学I 课后习题考试真题解答.pdfVIP

Advanced Microeconomic Theory I/A Fall 2009 Problem Set 1 Due Date: Oct 14,15, 2009 1. Let N = f1; :::; ng denote a set of n goods. For any x = (x ; x ; :::; x ) 2 1 2 n A n , deÖne + i (x) = fi 2 f1; :::; ng s.t. x x 8j =6 ig : i j (For simplicity assume that xi =6 xj for all i; j 2 N so that i (x) is unique.) For any x; y 2 A, consumer B weakly prefers x to y i§ xi(x) + xi(y) yi(x) + yi(y) 0: For example: Suppose x = (2; 2; 3) and y = (1; 7; 6). Then i (x) = 3 and i (y) = 2. In this case, Consumer B prefers y to x because (3 + 2) (6 + 7) 0. Is Consumer Bís preferences transitive? If so prove it. If no, give a counter example. 2. For any rational preference relation deÖned over some Önite set X , we can deÖne a preference relation r over the power set of X as follows: For any A and B , both subsets of X , we say that ArB if and only if the best element in A is weakly better than the best element in B according to . It is straightforward to see that r will possess the following properties: 1. Completeness, 2. Transitivity and 3. For any A and B , A B ) BrA. (The last condition means that if A is a subset of B , then B is always weakly better than A.) Is the converse also true? That is, suppose we have some preference relation r deÖned over the power set of X that is complete, transitive, and satisÖes condition 3, can we always come up with