《高级微观经济理论》习题解答 第一章,消费者理论

1、高级微观经济理论习题解答 第一章，消费者理论EX1.6不满足偏好是凸的的消费者偏好如“生病”和“旅游”（具体的图像自己画就知道了）EX1.7-1.10（略）EX1.11 PROOF:仅证A is closed set .由极限的定义： 宗上， A is a closed set.EX1.15 PROOF：B=x| px<=yis a compact and convex set: (1) B is a compact set: (2) B is a convex set:for all EX1.17 proof: (a) if u(x) is contionuous ,quasiconc

2、ave and increasing on X. B =x| px<=y then UMP will have a unique solution under the constrained condition Proof: we have known that（ EX1.15） B is a compact set , from Weierstrass THEOREM : . hence the solution of UMP exists in B and is unique in B.REMARK : Weierstrass THEOREM 可以参见任何有关数学分析的实数理论的章节

3、，这里就再赘述了。EX1.25 the answer to this question : . Marshallian demands : .EX1.28 sol u(X)= Because u is uniformly convergent , then structure Lagrangian function : .& in .We will have . Hence is the solution of this UMP.The maximum level of ulility value is .EX1.33 proof: From the definition of exp

4、enditure function (suppose that P be fixed)then s.t. u(X)>=u. the solution of this EMP is .then min. Beacause u(X) is continuous and increasing respect to X, so under the Axiosm of Locally Nonsatiation : .EX1.35 proof: （a）another approach to prove Roys identity:可以用WARP来分析（略） （b）；(c) (d) EX1.36思路

5、同上（略）EX1.42 proof：if is negative semidefinite matrix ,then its own-substitution terms is nonpositve: .Then When EX1.44(略，很简单，运用Slutsky Equation)EX1.46 proof: u(x) is liner homogeneous function (a) e(p,u)=min px s.t. u(x)>=u;e(p,1)=min px s.t. u(x)>=1.Because u(x) is homogeneous of degree 1 and

6、 linear function, u(x)=axSo u(ux)=uu(x)Furthermore u(x)>=1 u(ux)>=u holds on for all u>0Hence e(p,u)=ue(p,1)(b)EX1.47 proof:From Roys identity :So Insert Hence EX1.48 sol:运用Cournot aggregation formula 得到。EX1.49（a） proof ：运用变上限积分求导公式和Roys identity 可得：（b） 不懂这个测度的含义，有谁知道请指教！EX1.54 本人才疏学浅只会作第一问

7、，其他的思考良久没有结果！（a） let because u( ) is strictly concave ,so is g(t).then g(t)<g(t0)+g(t0)(t-t0) where t>t0 and there follows: From lagrangian theorem : s.t. Because and s.t. Then let where the rest number is 0.Then ,however Hence EX1.56 (a) (b) EX1.62 a=-8,P=32,b=2EX1.63 (a) right (b)right (c) wrong EX1.64 有关效用函数是位似的情况（homothetic）If from Eulers theorem :UMP:max u(x) s.t. px<=yStructure lagrangian function :FONC:因而，收入的边际效用与y无关由对偶性可得e(p,u)=ue(p,1)h(x)=uh(p,1)x(p,y)=yx(p,1)remark :由于我非经济学专业，在证明中文字较少，请读者见谅。同时由于知识的缺乏，我只是一个本科