高微高级微观经济学交大上海交大经济学.docx

1、高微高级微观经济学交大上海交大经济学1. 假设消费者生活两期，第一期消费者劳动获得收入，用来满足第二期的消费和储蓄，消费者在第二起不劳动，用储蓄来满足消费。假设消费者在第一期的消费为C1，储蓄为S，劳动收入为w，消费者在第二期的消费为C2，假设市场利率为r，贴现因子为01。设消费者的效用函数为：A representative consumer leave for two periods, in first period, he works and gets income for consumption and saving. In second period he doesnt work,

2、and uses his saving in first period to finance the consumption in second period. Assume his consumption in first period is C1, saving is S, income is w, consumption in second period is C2, the market interest is r, and discount factor is 01. The utility function is shown as follow:（1） 写出消费者的效用最大化问题（

3、2） 求出消费者的储蓄函数，并讨论利率的改变和储蓄的关系。(1) Try to write down the consumers utility maximization problems(2) Calculate out the saving function and discuss the relationship between interest rate and saving解：（1）消费者的效用最大化问题为： The utility maximization problem is shown as follow其中（3） 把约束条件带入目标函数中，有：substitute the b

4、inding condition in to the utility function we have： 对目标函数式关于C1求导，并令导数等于0Differentiate both sides with regard to C1, making first order derivative equal to 0解得对U(C1)求导得到U（C1）0因为S（r）=w-C1=上式对r求导得到：2. 一个有垄断势力的厂商面临需求曲线为（A为投入的广告费）：The demand function for a monopolist is shown as follow:c成本函数为：Its cost f

5、unction is shown as:(1) 试求出厂商利润最大化的A,Q和P的值Calculate out the A,Q, and P to maximize the profit(2) 试求出厂商在利润最大化时的勒那指数Calculate out the Lerner Index when the monopolist maximizes its profitSolution:（1）We have:, ,(3) Lerner index =P-MC/P=1/|=|dP/dQ*(Q/P)|=9/353. 考虑对电信业的需求，通常这种需求中存在着网络外部性，即随着上网或者电话客户用书上升，

6、消费者对电信服务的效用评价也会上升。下列效用函数就是反应“网络外部性”的：The demand for the Telecom usually involves externality, which is when the number of people using the same net increases, the consumers will benefit more. When the externality exists, the utility function is shown as follow:其中UX是消费者X的效用，n是网络中消费者的人数，x为消费者在电信客户中的先后

7、顺序，对n对x标准化后，我们有0n1,0x1，x越趋近于0，表示消费者越早成为电信用户。UX is consumers utility, n is the number of people using the same net, x represents the consumer is the xth person who starts to use the service. After normalizing n and x, we have 0n1,0x1, the more x is close to 0, the earlier the consumer starts to use t

8、he telecom service.Prove that with the externality, monopolist cant gain 100% market share.Proof:Then, we consider a situation where a consumer is indifferent between buy and not buy. We have：0=n(1-x)-PBecause x represents consumer x is the xth person who starts to use the service, we have x=n. Then

9、 we can get the equation for the marginal consumer:0=x(1-x)-pThe monopolist tries to maximize its profit, we have:Substitute the binding condition into the profit maximization function, we haveDifferentiate the maximization function with regard to x, we have the first order condition shown as follow

10、:2x-3x2=0X=2/3Now we have shown monopolist cant get 100% market share4消费者的效用函数如下： s-p if buyU=1 if not buy是消费者对商品的边际评价，s是商品质量，商品可以是高质量，也他可以是低质量。高质量商品S=1，低质量商品S=0。商家提供高质量商品的成本为C1，提供低质量商品的成本为C2，其中C2C1。因为C1，因此提供高质量商品是有效率的。is the marginal utility of the goods, s is the quality of the goods. The quality

11、can be high, which is S=1,or low, which is S=0. The cost of producing high quality goods is C1,and the cost of producing low quality goods is C0, C0C1. C1, High quality good is socially efficient.（1） 如果所有消费者都完全不知道商品质量，该市场的均衡将如何？If all the buyer are uninformed consumer, what is the market equilibrium

12、?（2） 如果有比例的消费者具有完全信息，1-的消费者完全不知道商品质量，市场的均衡将如何？If there are a informed buyers which accounts for of the total buyers, and the rest of buyers are uninformed, what is the market equilibrium?Solution:(1) H LBUY(-P, P-C1)(-P,P-C0)NOT BUY(0,0)(0,0)从博弈矩阵易知，最后的结果是消费者不购买商品，而商家只会提供低质量的商品。Based on the matrix,

13、we know that the market equilibrium is that buys wont buy the good and seller will only provide the low quality good.(2)在有具备完全信息的消费者的情况下，商家提供高质量商品和低质量商品的得益如下：The profits of producing high quality good and low quality good are shown as follow1=P-C1，如果商家提供高质量商品0=（1-）（P-C0）如果商家提供低质量商品H LBUY(-P, P-C1)(-

14、P,(1-)P-C0)NOT BUY(P-C1),0)(0,0)从博弈矩阵，我们得到，当P-C1（1-）（P-C0）时，垄断厂商会提供高质量的商品，最终的博弈结果是，商家提供高质量商品，而两类买方都会购买商品。Monopolist will supply the high quality good, when the P is sufficiently high.当P-C1（1-）（P-C0）此时，垄断厂商会选择以一定概率提供高质量商品，而不具备完全信息的消费者以一定概率购买商品。假设垄断厂商提供高质量商品的概率为r，提供低质量商品的概率为1-r，垄断厂商是的消费者在选择购买与不购买之间误差别

15、：r(-P)+(1-r)(-p)=0 得到 r=P/消费者以概率购买商品，以1-不购买商品，使得垄断厂商在提供高质量与提供低质量的商品之间无差异。（P-C1）+(1-)（P-C1）=（1-）（P-C0）可解除5. 某一市场需求曲线如下：The demand function is shown as follow:P=100-0.5（q1+q2）在该市场上，只有两家厂商，他们各自的成本函数为：There are 2 firms with the cost functions shown respectively as follow:C1=5q1，C2=5q2（1） 在斯塔克伯格模型中，谁会成为追

16、随者？In Stackelberg competition, which firm will be the follower?（2） 该市场最后的结局是什么？为什么？What is the market equilibrium? And why?解：（1）如果厂商1是领导者，厂商2是追随者: if firm 1 is leader and firm 2 is follower:P=100-0.5(q1*+q2)2=100-0.5(q1*+q2) q2-0.5 q22解得：q2=50-0.25 q1*将其带入厂商1的利润最大化目标函数中，得到：1=100-0.5(q1+50-0.25q1) q1

17、-5 q1解得1=3266，2=711以同样方法计算厂商2为领导时，两个厂商的利润。1=3003，2=918如果是古诺均衡，那么1=3200，2=900可以画出博弈矩阵：厂商2厂商1领导者leader追随者follower领导者leader(3200, 900)(3266,711)追随者follower(3003,918)(3200,900)可知，在有限次博弈中，最终结果是古诺均衡。但在无限重复博弈中，厂商可采取以下策略：对厂商1而言，他会选择如下策略：加入厂商2率先宣布自己的产量为q2，那么厂商1可以选择这样一个产量q1，使得厂商2的利润为负，同时是自己的利润为正。那么这意味着下面两个不等式

18、成立：According to the matrix, we know that in finite repeated game, the market yields Cournot equilibrium, but in the infinite repeated game, firm 1 can take the following strategy: if firm 2 declare its output, firm 1 could choose to produce q1(which is may not a profit maximization output) to make t

19、he profit of firm 2 less than 0 and keep its own profit positive. So, the following 2 conditions should be satisfied:（100-0.5q1-0.5q2）q1-5q10（100-0.5q1-0.5q2）q2-0.5q220这意味着着q210，否则q1可以选择一个产量使得自己盈利，这样给定厂商2的产量q2，厂商1的目标是：We can derive from the two inequalities, that q210, if q210, firm 1 could take act

20、ion to make firm 2s profit negative. The profit maximization function for firm1 is shown as followMax:（100-0.5q1-0.5q2）q1-5q1一阶条件解得q1=95-0.5q2厂商2的利润最大化问题可以写为：The firm2s profit maximization function is shown as follow:（100-0.5（95-0.5q2）-0.5q2）q2-0.5q220q210如此，得到厂商二利润最大化时的收益为450，还不如做追随者时的711高。Now we g

21、et that firm 2s profit(s.t. 0q210) is 450, which is less than 711.厂商1当领导者时，厂商2有可以采用同样策略来威胁企业1，为了是企业2策略无效，厂商1不能生产超过180的产量，此时他的利润仍然为3266。With the same method, we know that firm 2 could do the same to firm 1. In order to make the strategy irrelevant, firm 1 can only produce maximum number of units of 1

22、80. However, its profit under this condition is still 3266. 因此，在无限重复博弈中，厂商1会成为领导者，厂商2会成为追随者。Therefore, in the infinite repeated game, firm1 will become the leader and firm2 will become the follower.（2）在无限重复博弈中，厂商1会成为领导者，厂商2会成为追随者；在有限次重复博弈和一次博弈中，两个厂商最终会形成古诺均衡。（理由如前所述）7 假定二手车市场上的质量分布服从0,2,但卖主与买主一样不知道产

23、品质量，如果卖主和买主的效用函数分别为U1=M+qn和U2=M+3/2qn，预算的约束分别为y1=M+Pn，和y2=M+Pn，请证明：市场均衡时有P0，q0，并不存在逆向选择。Assume the quality of used car is uniformly distributed at0,2, but the sellers dont know the quality of the product as well. If the utility functions of seller and buyer are shown respectively as follow U1=M+qn a

24、nd U2=M+3/2qn. The budget constraint is shown as y1=M+Pn and y2=M+PnProve: In equilibrium, P0, q0, and adverse selection doesnt exist.Proof: 证明：在双方都不知道二手车的质量的情况下，买主的效用函数与以前一样：EU2=y2+3/2-Pn要买主买车，必须有3/2P卖主的效用函数为：EU1=y2+-Pn要卖主卖车，必须有P因此，我们可以退出，P3/2，且市场上所有车都会被出售。8. 一个消费者的初始收入为I， 如果发生事故，他会损失L，设出现事故的概率为，不发

25、生事故时，不会有损失。消费者是严格厌恶风险的，保险公司是风险中立者，他要求消费者在投保时支付保费为P，但保证在发生事故时赔偿S。Assume the initial income for a representative consumer is I, if there is an accident, he will lose L. we assume the probability of the accident is , if the accident does not happen, he will lose nothing. The consumer is risk averse and

26、 the insurance company is risk neutral. The company charges P for the insurance and guarantees to cover S if the accident happens.（1） 写出投保人的期望效用，写出保险公司的期望利润。Write down the expected utility function for the consumer and the expected profit for the insurance company.（2） 如果保险公司要选择S与P，来确保最大化保险公司的期望利润，应该

27、满足投保人怎样的“约束”If the insurance company have to choose S and P to maximize its profit, which constraint should it satisfy?（3） 解（2）中列出的数学规则，证明最优赔偿金S应该满足S=LBased on the second question, prove that the S that maximize the expected profit equals to L.解：（1）假设投保人的效用函数为U(X),这个效用函数呈现凹性，即V0，V0,则他的期望效用为：EU(X)=U（

28、I-P-L+S）+(1-)U（I-P）保险公司的期望利润为：E=（P-S）+(1-)P=P-S（2）如果保险公司的目标是最求利润最大化，他么他的保险计划至少要使得投保人的境况和不投保时一样好。 EU（P,S）EU（0,0）具体化以后为：U（I-P-L+S）+(1-)U(I-P)U（I-L）+(1-)U(I)整理得：U(I-P-L-S)-U(I-L)+(1-)U(I-P)-U(I)0（3） 保险公司的利润最大化问题为：建立拉格朗日函数求解：F.O.C因为U（.）0,因此U(.)严格递减，所以I-P+S-L=I-P从而得出S=L9（This is the last homework, so I t

29、hink you guys already have the English version）生产一种公共物品P和一种私人物品G的经济生产可能性边界由下式决定：并且，该经济由100个完全相同的个人组成，每个人有如下形式的效用函数：Gi是个人在私人商品生产中的份额（=G/100）（1） 如果G和P的市场是完全竞争的，将会生产出多少？在此情况下典型的个人效用会如何？（2） G和P的最优生产水平如何？典型的消费者效用水平如何？应该怎样对商品进行征税已达到这一结果？解：（1）如果G和P的市场是完全竞争的，在均衡时必有以下条件：MRSP,G=MRTP,G得到：因为Gi=G/100,所以我们解得：G=100P把G=100P带入约束条件中，我们得到P=0.704，Gi=0.704(2)按照萨缪尔森加总规则计算G和P的最优生产水平：解得G=10P，将其带入生产可能性边界中，解得P=5，G=50, Gi=0.5消费者的效用为U=0.5*101/2=1.58政府可以通过向消费者消费公共品征收从量税，是的在竞争条件下公共物品的供给达到有效数量，假设私人品的的价格为1，公共物品的价格为t（所征税收）。在达到竞争性均衡时，每个消费者的边际替代率相同，都等于私人物品和公共物品的价格之比，即：