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1、condition. Becoming a rich country also depends on other factors such as political system, social infrastructures, and culture. Additionally, no example of a country can be found so far that it is rich in the long run, that is not a market economy. A positive statement state facts while normative st
2、atement give opinions or value judgments. Distinguishing these two statements can void many unnecessary debates. 1.2Language and Methods of Mathematics This section reviews some basic mathematics results such as: continuity and concavity of functions, Separating Hyperplane Theorem, optimization, cor
3、respondences (point to set mappings), fi xed point theorems, KKM lemma, maximum theorem, etc, which will be used to prove some results in the lecture notes. For good references about the materials discussed in this section, see appendixes in Hildenbrand and Kirman (1988), Mas-Colell (1995), and Vari
4、an (1992). 1.2.1Functions Let X and Y be two subsets of Euclidian spaces. In this text, vector inequalities, =, , and , are defi ned as follows: Let a,b Rn. Then a = b means as= bsfor all s = 1,.,n; a b means a = b but a = b; a b means as bsfor all s = 1,.,n. Defi nition 1.2.1 A function f : X R is
5、said to be continuous if at point x0 X, lim xx0 f(x) = f(x0), or equivalently, for any 0, there is a 0 such that for any x X satisfying |x x0| 0 such that for any x X satisfying |x x0| 0 0if x = 0. The correspondence is closed but not upper hemi-continuous. Also, defi ne F : R+ 2Rby F(x) = (0,1). Th
6、en F is upper hemi-continuous but not closed. Figure 1.2 shows the correspondence is upper hemi-continuous, but not lower hemi- continuous. To see why it is upper hemi-continuous, imagine an open interval U that encompasses F(x). Now consider moving a little to the left of x to a point x. Clearly F(
7、x) = y is in the interval. Similarly, if we move to a point xa little to the right of x, then F(x) will inside the interval so long as x is suffi ciently close to x. So it is upper hemi-continuous. On the other hand, the correspondence it not lower hemi-continuous. To see this, consider the point y
8、F(x), and let U be a very small interval around y that 23 does not include y. If we take any open set N(x) containing x, then it will contain some point xto the left of x. But then F(x) = y will contain no points near y, i.e., it will not interest U. Figure 1.2:A correspondence that is upper hemi-co
9、ntinuous, but not lower hemi- continuous. Figure 1.3 shows the correspondence is lower hemi-continuous, but not upper hemi- continuous. To see why it is lower hemi-continuous. For any 0 5 x5 x, note that F(x) = y. Let xn= x 1/n and let yn= y. Then xn 0 for suffi ciently large n, xn x, yn y, and yn F
10、(xn) = y. So it is lower hemi-continuous. It is clearly lower hemi-continuous for xi x. Thus, it is lower hemi-continuous on X. On the other hand, the correspondence it not upper hemi-continuous. If we start at x by noting that F(x) = y, and make a small move to the right to a point x, then F(x) sud
11、denly contains may points that are not close to y. So this correspondence fails to be upper hemi-continuous. Combining upper and lower hemi-continuity, we can defi ne the continuity of a corre- spondence. Defi nition 1.2.11 A correspondence F : X 2Yat x X is said to be continuous if it is both upper
12、 hemi-continuous and lower hemi-continuous at x X. A correspondence F : X 2Yis said to be continuous if it is both upper hemi-continuous and lower hemi-continuous. 24 Figure 1.3:A correspondence that is lower hemi-continuous, but not upper hemi- continuous. Remark 1.2.10 As it turns out, the notions
13、 of upper and hemi-continuous correspon- dence both reduce to the standard notion of continuity for a function if F() is a single- valued correspondence, i.e., a function. That is, F() is a single-valued upper (or lower) hemi-continuous correspondence if and only if it is a continuous function. Defi
14、 nition 1.2.12 A correspondence F : X 2Ysaid to be open if ifs graph Gr(F) = (x,y) X Y : y F(x) is open. Defi nition 1.2.13 A correspondence F : X 2Ysaid to have upper open sections if F(x) is open for all x X. A correspondence F : X 2Ysaid to have lower open sections if its inverse set F1(y) = x X
15、: y F(x) is open. Remark 1.2.11 If a correspondence F : X 2Yhas an open graph, then it has upper and lower open sections. If a correspondence F : X 2Yhas lower open sections, then it must be lower hemi-continuous. 1.2.7Continuity of a Maximum In many places, we need to check if an optimal solution i
16、s continuous in parameters, say, to check the continuity of the demand function. We can apply the so-called Maximum Theorem. 25 Theorem 1.2.6 (Bergs Maximum Theorem) Suppose f(x,a) is a continuous func- tion mapping from A X R, and the constraint set F : A X is a continuous correspondence with non-empty compact values. Then, the optimal valued function (also called marginal functio
