课程名称︰数量方法入门 课程性质︰经研所必修 课程教师︰朱玉琦 开课学院：社科院 开课系所︰经济所 考试日期（年月日）︰110.08.30 考试时限（分钟）：180 试题 : 部分数学式以latex语法呈现。 Preamble: You have 3 hours to answer the following questions. The total number of points is 110. You need to answer each questions in English. If you have stated a theorem in lecture notes (including homework), you may use it without proving it unless I explicitly ask you to, but you need to describe what the theorem is and why you can apply that theorem. For example, if there are some assumptions for that theorem to be applicable, you need to show that those assumptions are met in your problem. If a theorem is famous, e.g. implicit function theorem, you can simply refer this theorem by its name. 1. (18 points) Let S = \Set{x\in\R | x=1-1/n, n\in\Z^+}\cup\Set{1}. Is S closed ? Is it comapct? Is it convex? 2. (8 points) Consider f : \R^n\to\R. Define U_f(\overline{v})=\Set{x | f(x)\geq\overline{v}} where \overline{v}\in\R . Show that U_f(\overline{v}) is a convex set for every \overline{v} if f(θx^a+(1-θ)x^b)\geqθf(x^a)+(1-θ)f(x^b) for all x^a and x^b and for all θ\in[0,1]. 3. (8 points) Consider f : \R^n\to\R^1 and \nabla f(x)\neq0 for all x. Let D be an open set in \R^n. SHow that the solution of \max{f(x)|x\in D} does not exist. 4. (6 points) Consider f : [a,b]\to\R, and its derivatives exist at every point in (a,b) and f continuous at both end points. Assume that f'(x) > 0 for all x \in (a,b). Prove that f(b) > f(a). 5. (12 points) Consider a function f : \R^3\to\R as follow: f(x1,x2,x3) = (x1x2+x3^2)/(x1^2+x2^2+2x3^2) if (x1,x2,x3)\neq(1,1,1) = 0 if (x1,x2,x3)=(1,1,1). (a) Show that f is not continuous at (1,1,1). (b) SHow that the 1st-order derivetive of f() wrt x3 does not exist at (1,1,1) 6. (10 points) Discuss why the following proofs are incorrect: (a) Consider f : S\subset\R\toR, where S is an open set in \R. Assume that f is continuous at a point c in S and that f(c) > 0. Prove that there exists an open set U \subset S such that for all x\in U, f(x) > 0. Proof: Because f(c) is an interior point of (0,\infty), there exists a \delta>0 such that B(f(c),\delta)\subset(0,\infty). Because B(f(c),\delta) is an open set in \R, its inverse image f^{-1}(B(f(c),\delta)) is open in S. Therefore, we find and open set U = f^{-1}(B(f(c),\delta)) in S such that for all x \in U, f(x)\in B(f(c),\delta)\subset(0,\infty), implying that f(x)>0. (b) Prove that if S_1 and S_2 are compact then S_1+S_2 is compact. Proof: Pick any sequence {z_n} and z_n\in S_1+S_2 for all n. We want to show that we can find a convergent subsequence of {z_n} such that its limit z is in S_1+S_2. Because z_n\in S_1+S_2, this implies that we can find a point x_n\in S_1 and y_n\in S_2 such that z_n=x_n+y_n. Because S_1 is compact, and {x_n} is a seque- nce in S_1, we can find a convergent subsequence of {x_n}, denoted by {x_{k(n)} such that \lim_{n\to\infty}x_{k(n)}=x\inS_1. Because S_2 is compact, and {y_n} is a sequence in S_2, this implies that its subsequence {y_{k(n)} converges to a a point y\in S_2. Therefore, we find a subsequence {z_{k(n)}} converges to z=x+y, and z\in S_1+S_2, which complete that proof. 7. (10 points) Consider a function Q=5K^{0.5}L^{0.5}. Currently, it is using the input bundle (K,L)=(1,1). It's clear that Q=5 when (K,L)=(1,1). (a) Use the first-order Taylor approximation to estimate the output when both inputs increase by 0.5 units. (b) Use the second-order Taylor approximation to estimate the output when both inputs increase by 0.5 units. 8. (20 points) Consider the following utility maximization problem. A consumer maximizes that \max_{x1,x2}U(x1,x2)=x1+a\log(x2), where a is a positive consta- nt number. The constraint set is D(p1,p2,I)=\Set{(x1,x2)\in\R^2 | p1x1+p2x2\leq I, x1 and x2\geq0}, where p1>0 and p2>0. In class, we have checked that the solution to the problem exists and the constraint qualification holds for all possible cases of solutions, and the derivatives exist, so the solution must satisfy the Kuhn-Tucker first-order co- nditions. Now answer the following questions: (a) (8 points) Set up the Lagrangian function and write down all the 1st-order conditions of Kuhn-Tucker. (b) (2 points) Prove that for all x1 > 0 and x2 = 0 cannot be the optimal solu- tion. You only have to check the Inada condition. (c) (10 points) Show that x1^*=0 and x2^*=I/p2 if I\leq ap1, and that x1^* = (I-ap1)?p1 and x2^* = ap2/p2 if I-ap1 > 0. --

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