课程名称︰数量方法入门 课程性质︰经研所硕一必修 课程教师︰王建强 开课学院：社会科学院 开课系所︰经研所 考试日期（年月日）︰2017/09/08 考试时限（分钟）：9:10~12:10 共3小时 试题 : Instruction: You have two hours. Write your answers clearly, with good penmanship and good syntax. A correct but unintelligible answer is a wrong answer. You are allow to use any theorem, lemma or property we have proved in class or in notes. If you want to apply any theorem or property we have never talked about, please prove it. 1. (10 points) Definition: Write down the definiton of a compact set. 2. (10 points) Definition: Let (X, dx), (Y, dy) be metric spaces, and f: X --> Y. Write down the definition: f is continous at x ﾑk X 3. (10 points) In a metric space, is an infinite intersection of open sets always open? If your answer is "Yes," prove it. If not, provide a counterexample. 4. (15 points) Prove the following: an infinite subset of a compact set K has a limit point in K. 5. (15 points) Prove that closed subsets of compact sets are compact. 6. (15 points) Prove that if a sequence in R^n is bounded then it has a covergent subsequence. 7. (a) (5 points) Give an example of two disjoint sets which cannot be seperated (b) (5 points) Give an example of two closed convex sets that are disjoint but cannot be strictly seperated 8. (10 points) Let E', E" be linearly independent sets of vectors in V. Show that E'∩ E" is linearly independent. 9. (a) (5 points) Find the eigenvalues and eigenvectors of A = ╭ ╮ │ 1 2 │ │ │ │ 3 2 │ ╰ ╯ (b) (5 points) Find the range and rull space of A = ╭ ╮ │ 1 2 3 │ │ │ │ 1 1 0 │ ╰ ╯ 10. (15 points) (Borrowing Constraint Model) Consider the following problem: A household wants to consume at both period 1 and period 2, but it can only work at period 2. Working generates h units of consumption goods with no cost or disutility. Although the household cannot work at period 1, it is allowed to borrow consumption goods from the credit market with interest rate r. The household can borrow up to L ≧ 0, and we assume that L is exogeneously determined. Solve the following optimization problem step by step. sup c1,c2 ∈ R ln(c1) + ln(c2) s.t. (1 + r)c1 + c2 = h c1 ≦ L c1 ≧ 0 c2 ≧ 0 (a) Rewrite the problem into the following form inf x ∈ X f(x) s.t. hi(x) ≦ 0 li(x) = 0 (b) Write down the Lagrangian. (c) Following Karush-Kuhn-Tucker Conditions, list the sufficient and necessary condition for optimal solutions (d) Argue that the constraints c1 ≧ 0 and c2 ≧ 0 are not binding (meaning that we must have c1 > 0 and c2 > 0) (e) Solve the optimal (c1*, c2*). Discuss that under what condition, the solution is corner; under what condition, the solution is interior? --

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