专业课资料 练习题 真题 中心习题 2011 FALL micro 雷晓燕

1 IntermediateIntermediateIntermediateIntermediateMicroeconomicsMicroeconomicsMicroeconomicsMicroeconomics FallFallFallFall2011201120112011 HandHandHandHandin in in inForm Form Form Form HandHandHandHandWritingWritingWritingWritingOnlyOnlyOnlyOnly AssignedAssignedAssignedAssignedTime Time Time Time 2011 9 152011 9 152011 9 152011 9 15 DueDueDueDueTime Time Time Time bybybyby2011 9 22 2011 9 22 2011 9 22 2011 9 22 5 40pm 5 40pm 5 40pm 5 40pm 1 Draw indifference curves over baseball and football to represent the following descriptions of consumer preferences a I don t care whether I watch football or baseball I just like to watch sports b I only watch football baseball is boring c I don t like to watch football it is too violent I don t like to watch baseball either it is too dull d I like to watch a little football and a little baseball more than a lot of either while I like sports I enjoy variety as well e My friend and I only like to watch sports together but since I like football and he likes baseball we always watch equal amounts of each f I like watching baseball and football although I enjoy baseball somewhat more than football Unfortunately after an hour of watchingTV my eyes get tired and I don t enjoy it anymore 2 Suppose that U x2y3 for this utility function a Is marginal utility decreasing b Is the marginal rate of substitution between x and y decreasing c Draw indifference curves for this utility function What properties do the underlying preferences obey 3 Connie Consumer has a monthly income of 200 which she allocates between two goods meat and potatoes a Suppose meat costs 4 per pound and potatoes cost 2 per pound Draw her budget constraint b Suppose also that her utility function is given by the equation U M P 2M P What combination of meat and potatoes should she buy to maximize her utility c Connie s supermarket has a special promotion If she buys 20 pounds of potatoes at 2 per pound she gets the next 10 pounds for free This offer applies only to the first 20 pounds she buys All potatoes in excess of the first 20 pounds excluding bonus potatoes are still 2 per pound Draw her budget constraint d An outbreak of potato rot raises the price of potatoes to 4 per pound The supermarket ends its promotion What does her budget constraint look like now What combination of meat and potatoes maximizes her utility 2 4 Let income be I 100 Px 1 Py 1 and utility U xy a Write the budget constraint and find the optimal consumption bundle for the consumer b Suppose that a tax of 0 5 is applied per unit consumed of good x Write the new budget constraint and solve for the optimal bundle Compute the revenue the government earns from the tax revenue tax units consumed Label this revenue R c Consider an alternative tax scheme of a lump sum income tax The amount of the tax is R What is the new budget constraint of the consumer What is the new optimal consumption bundle d Does the optimal bundle of part b satisfy the budget constraint of part c e Which tax scheme b or c do consumers prefer AnswerAnswerAnswerAnswertotototoPS1PS1PS1PS1 1 a This means that football and baseball are perfect substitutes that is that consumer satisfaction depends on the total quantity of sports watched not on which specific sport is being watched e The two programs are only watched in a proportion of one to one The indifference curves are right angled with the angle lying on a ray of slope 1 go through the origin 2 3 a b Connie s budget constraint isIt is easy to see that her bundles will be c d Connie will maximize her utility only by spending all her money on meat it is easy to verify Solution isM 50 P 0 4 a The budget constraint is 100 x y To find the optimal consumption bundle we set the slope of the indifference curve equal to the slope of the budget constraint that is y x 1 so that y x and use the equation in the budget constraint Hence 100 x x so that x 50 and y 50 b The new budget constrain is 100 1 5x y Since the slope of the budget constraint is now 1 5 we have y x 1 5 that is y 1 5x Plugging this back into the budget constraint gives us 100 1 5x 1 5x Hence x 100 3 and y 1 5x 50 The revenue from the tax is equal to R 0 5x 50 3 c The budget constraint is now 100 R x y Since the slope of the budget constraint has not changed we still have x y Plugging this back into the budget constraint gives us 100 R 2x or x 100 R 2 Since R 50 3 we have x 250 6 and y 250 6 d The quantities consumed in part a satisfy the budget constraint in part c since total expenditures are equal to 100 3 50 250 3 and the total available revenue is also 100 R 250 3 e the argument in d shows that the bundle consumed in c is revealed preferred to the bundle consumed in b the consumer could have chosen the b bundle but he chose another one instead Therefore consumers prefer the tax scheme in c 2011 年秋季 中级微观经济学 PS2 上交时间 10 月 8 日课前 1 某消费者的效用函数为 2 1212 U x xxx 1 x 为商品 1 的数量 2 x 为商品 2 的 数量 商品 1 和 2 的价格分别为 1 p 2 p 消费者的收入为m a 求商品 1 和 2 的马歇尔需求函数 补偿需求函数 间接效用函数 支出函数 b 如果 m 24 1 p 1 2 p 1 现在 2 p 上升为 2 求此消费者关于商品 1 价格变 动的收入效应和替代效应所带来的消费量的变化 c 根据 a 中的结果验证恩格尔加总规则 2 假设你的效用函数为 u x y x 的价格为 y 的价格为 收入为 请计 算 上升时 x消费量变化的收入效应和替代效应各是多少 请注意分类讨论 假设消费者在价格变化前收入全部用来消费 x 3 某北大学生国庆期间依靠公共汽车 b 火车 t 或者飞机 p 旅行 他旅 行的效用函数为 Ub tp b t p 分别代表各种旅行方式的里程数 假设火 车旅行与公共汽车旅行的价格比 tb pp 固定不变 a 请定义陆路运输的组合商品 g b 他要在陆地运输工具 g 与航空运输工具 p 中做出选择 写出其最优化问 题并求出他对 g 与 p 的需求函数 c 一旦该学生确定了花费在 g 上的货币额 他会怎样将这些货币在 b 与 t 之间进 行分配呢 4 请证明 对于一个将所有收入全部用来消费 1 x 和 2 x 两种商品的消费者而言 1 x 和 2 x 不可能都是劣等品 PS2 答案 1 a 写出最优化问题 2 1212 Max xU x xx 1 122 st p xp xm 应用拉格朗日方法 2 12121 122 x L x xxp xp xm 利用 F O C 条件求出 12 12 21 33 mm xx pp 将上面两式的马歇尔需求函数代入原效用函数 得到间接效用函数 3 12 2 12 4 27 m V p p m p p 将上式间接效用函数中的 V 换为既定效用 u m 换为支出 E 于是可以求得支出函数 22 1212 33 12 27 3 44 up pup p E p p u 对上式应用谢泼德定理得到补偿需求函数 2 3 112 1 2u c p xp p u p 2 1 3 212 2 2 up 4 c xp p u p b 先求出价格变化前后的消费者选择 0 1 x 16 1 1 x 16 于是价格变化前消费者得到的效用为 2 0 u 168 2048 价格变化后 消费者要保持之前的效用需要消费 x1的量可以应用补偿需求函数求出 112 c 023 3 3 1 1 2u2 168 2 x 162 1 p p 于是对于商品 1 的价格变化的替代效应和收入效应所引起的消费量的变化分别为 c 1 x 1 3 162 16 I 1 x 16 1 3 162 总效应为 0 c 恩格尔加总规则表明一个人所买所有商品的收入弹性的加权平均为 1 将 a 中的结果验 证即可 比较简单 2 开始时 所有的收入用来购买 x 根据消费者的效用函数形式 我们可以推断 现在 x 的价格由 变化为 a 所有的收入仍然用来购买 x1 0 b 0 c X Y X Y Y X IC IC 3 4 该消费者的预算约束式为 xy p xp yI 两边对收入I求导 可得 1 xy xy pp II 若 x 和 y 均为劣质品 则 x I 0 y I thus1ab d We know that ln ln d KLRTSKL d RTSKLRTS and 11 a bab LK RTSMP MPAbK LAaKLbK aL thus ln ln ln 1 ln ln ln ln K LK LK L RTSbK aLK Lb a or you may use 1 d K LRTSd K LbK aLa b d RTSK Ld bK aLK Lb a See it has nothing to do with either a or b e Here comes the micro foundation for Solow residual As I said in theTAclass first we take a log transformation of both sides of the production function lnln lnlnQA taK bL then lnln lnlnQA tKL ab tttt and this tells you QAKL GGaGbG thus AQKL GGaGbG So 2 5 0 3 1 75 0 7 1 1 25 A G Given wining the bet only requires 1 A G the Weasleys twins will win For more information of Solow residual you may refer to P201 P202 in our textbook Chapter 8 the English version 2 Answer 3 a Min16vkwlr St 0 2 50 2 5 4klq FOC 0 250 75 lkv 1 0 250 75 klw 2 we get 2 16 qw k v 0 25 0 25 4klq 3 2 16 qv l w Thus 2220 5 161616 16168 qwqvq wv C v w r qkvwlrvwrr vw Or you may use what I mentioned in theTAclass since it s a Cobb Douglas thus the proportions of money spent on k andlis a and bif a b qk l so the money spent on k andlis the same you just need to calculate kv and C 2kv 16r b Max 0 25 0 25 4 16pf k lC qpklkv wlr FOC 0 250 75 plkv 4 we get 2 0 51 5 p k wv 0 250 75 pklw 5 2 1 50 5 p l wv Thus given v 3 w 12 r 10 and p 60 we get 0 25 0 250 250 250 250 50 25 0 250 75 44 60 44 4 4 40 3 12 vvvpp qklkkk wwwwvwv 2222 0 25 0 51 50 51 50 51 5 2 4 1616 vppvpp p v w rpvwrr wwvwvw wvwv 2 2 60 60 3 12 10 16 10400 3 12 c When v 3 w 12 r 10 and p 60 we can see 2 1 50 5 603600 50 12312 12 3 l If v 3 w 27 r 10 and p 60 44 60 26 67 27 3 p q wv 22 1 50 5 60 14 8 27 27 3 p l w v The substitution effect can be derived using result from a given q 40 22 403 33 33 161627 qv l w Thus the substitution effect is 50 33 33 16 67 while the output effect is of course 33 33 14 8 18 53 4 a Max 2 20 25 qq FOC2q 20so q 10 2 20 10 1025 75 b producer surplus 25 cos 7525100fixedt c You can useMax 2 25 25 qq FOC2q 25so q 12 5 producer surplus 2 2525 12 5 12 525 25156 25 or you may use SMC 2q producer surplus 25 25 2 20 20 10010010056 25156 25 24 pp dp d SMC 2q p SMC so q 2 p producer surplus 2 2 2525 224 ppp p 中级微观经济学 1 班 PS4 Deadline 11 月 17 日课前 1 在一个纯交换的完全竞争市场上有两个消费者 A 和 B 两种商品 x 和 y A 的初始禀赋 xA yA 4 5 B 的初始禀赋 xB yB 8 3 他们的效用函数分别为 U xA yA xAyA U xB yB xByB a 求市场均衡时的相对价格和各消费者的消费量 b 给出帕累托最优分配的契约线的表达式 c 如果 A 的效用函数为 U xA yA min xA yA 求一般均衡时的相对价格和契约线 2 假设小陈一个人在果发园过着世外桃源的生活 独自生产并消费食物 F 和衣服 C 假设某个月 她决定工作 200 个小时 至于到底把这些时间用到捕鱼还是收椰子上是无 差别的 小陈的食物产量为F f l 她的衣服产量为 c Cl 其中 f l与 c l分别为花 在食物和衣服生产上的劳动时间 并且200 cf ll 小陈的效用函数为UF C a 如果小陈不与外面世界贸易 她将如何配置他的劳动 F 和 C 的最优水平是多少 她的 效用是多少 食物替代衣服的 RPT 是多少 b 假设可以进行贸易 且小陈能以2 f c p p 的价格比率进行交易 如果她仍然按照 a 中 的产量生产食物与衣服 在给定上述贸易机会的情况下 她会做出什么样的消费选择 她此时的效用水平是多少 c 如果小陈调整她的生产以利用世界价格的优势 b 的答案会怎么变化 d 请用图形表示 a b c 中的结果 3 假设在北京和德州之间每天有且只有一班火车通行 该火车有 150 个座位 不考虑站票 等情况 最多容纳 150 名乘客 运行一天的成本是 2500 元 不管有多少乘客乘坐 每 天乘该列车的需求函数为 40 5PQ a 请画出每天乘车的需求曲线和边际成本曲线 b 假设一个私有公司垄断该火车线路的运行 并且只能定一个价格 请问价格会定在什么 水平 c 求社会最优的定价应该是多少 d 假设 b 问中的私有公司可以进行一级价格歧视 请问它会如何定价 这个定价是社会最 优水平吗 4 假设 A 是市场上唯一的出租计算机的企业 假设它有两类潜在的顾客 100 家企业和 100 家学术机构 每一个企业客户和学术机构的需求函数分别为 Q 10 p 和 Q 8 p p 为租赁一台计算机的单价费用 A 的边际成本恒为 0 租费为向 A 企业租赁计算机支付 的门槛费用 一次性支付 使用费为支付门槛费用后租用每台计算机的单价费用 a 假设 A 能够区分两类客户 它对企业和学术机构收取的租费和使用费各是多少 b 假设不能区分两类客户 且不收取租费 那么它向两类客户收取的统一使用费是多少 c 假设不能区分两类客户 只能收取统一的租费和使用费 那么它设定的使用费和租费各 为多少 利润是多少 d 假设只有 10 家学术机构 重做第 3 问 PS4 参考答案 1 a 令x的价格为 p y 的价格为 1 求A的需求函数 AAAAA Lx yIpxy 其中 45 A Ip 22 AA AA II F OCxy p 即 55 2 2 22 AA xyp p 同理 BBBBB Lx yIpxy 其中 83 B Ip 22 BB BA II F OCxy p 即 33 4 4 22 BB xyp p 4583 481212 22 4223252525 612 34466 AB ABAB pp xx pp pxxyy p b max AA x y 12 8 AA stxyU 12 8 AAAA Lx yxyU F OC 8 0 AA A L yy x 12 0 AA A L xx y 8 12 AA AA yy xx 2 128 012 3 AAAAAAAAA yx yxx yyxx c 令x的价格为 p y 的价格为 1 求 A 的需求函数 45 11 A AAAA Ip xyxy pp 同理 BBBBB Lx yIpxy 其中 83 B Ip 22 BB BA II F OCxy p 即 33 4 4 22 BB xyp p 453 53848 12 1053 16 AB p yyp p p 此时的契约曲线为 08 8 812 AAAAAAAAA xyyxxxyyx 2 d 3 a b MR MC 垄 断 价 格 会 定 在 P 20 的 水 平 上 但 是 此 时 垄 断 企 业 其 实 得 到 负 利 润 20 100 2500 500 所以垄断企业会停止火车运行 c 社会最优定价需使社会总福利最大 于是价格可以定在 P 0 10 但此时由于企业利润 为负 所以需要给垄断企业以补贴才能使它运营火车 或者就是政府直接接管该企业运营火 车 d 如果垄断企业可以实行一级价格歧视 则其会对每个消费者收取能接受的最高价格 即沿着 需求曲线区别定价 赚取所有的剩余 直至 Q 150 是社会最优水平 4 a A 能够区分两类客户 且收取租费和使用费 对企业的租费为 50 对学术机构的租费为 32 使用费都为 0 b A 不能区分两类客户 只收取统一的使用费 则 A 的最优化问题为 若 0 p 8 时 Max 100p 10 p 100p 8 p p 4 5 ok 若 8 p 10 时 Max100p 10 p p 5 wrong 所以 p 4 5 c A 不能区分两类客户 只收取统一的租费 r 和使用费 p 假设 A 使两类客户都参与交易 Max200r p Q1 Q2 200 0 5 8 p 2 100p 10 p 100p 8 p 于是可以得到 p 1 于是 r 49 2 利润为 6500 假设 A 舍弃一类客户 根据需求函数 我们可以知道 只可能是舍弃学术类机构 那么这时 A 所能赚取的最大利润为 100 50 5000 r 50 p 0 小于 6500 所以 A 会选择 p 1 r 49 2 利润为 6500 两类客户都参加交易 d 由第三问中的类似分析 可以知道 此时 A 会舍弃学术类机构 将 r 定为 50 p 定为 0 利润为 5000 P Problem roblem S Setet 5 5 1 1 Wall at Silicon ValleyWall at Silicon Valley Jobs and Gates produce personal computers Each one has a cost function given by The market demand is where a If Jobs and Gates act to maximize their own profits taking his rival s output as given what will be the Cournot equilibrium for each one What are the profits for each one b One day Gates finds that it will be much better if they cooperate The two collude and behave like a Cartel What will be the profit maximize output and equilibrium price What are the output and profit for each one c An apple hit Jobs when he is reading under the tree suddenly he comes up with a wonderful idea Jobs new product Marc can be produced in accord with the new cost function Jobs dominates the market and become the market leader After he announces his output decision Gates decides his own output to maximize his own profit What are the output and profit for Jobs and Gates 2 2 Petroleum CompetitionPetroleum Competition Sinopec and China petro are selling petroleum They compete by choosing price Their demand functions are and respectively Both of their marginal cost equal zero There is no fixed cost a Suppose the two set their own price at the same time Find the Nash Equilibrium price and calculate the profit for each corporate b Suppose that Sinopec chooses its price first and then China petro sets its price what will the equilibrium prices be this time Calculate the profit for each firm Is there a first mover advantage for Sinopec 3 3 Battle of the MarneBattle of the Marne At the Battle of the Marne France has 3 choices F W R meaning fight wait retreat Meanwhile Germany also has 3 choices F W R France and Germany move at the same time Their payoff matrix is as follows Please find all Nash Equilibriums Hint Don t forget to delete the dominated strategy as I mentioned at the TA class 4 4 DilemmaDilemma at Downton Abbey at Downton Abbey Matthew Crawley came to Downton village He will be the next Earl of Grantham if the Mrs Earl s unborn baby is not a boy Mathew wants to propose to Lady Marry Crawley the daughter of current Earl Although Marry loves Matthew she knows clearly that if she marries to Mathew and the baby is a boy Matthew will have to go back to Manchester and she will be the wife of a common solicitor and lose the identity of a gentry However if the baby is a girl Matthew will be the Earl and she will be the next Mrs Earl If she doesn t marry to Mathew and the baby turns to be a boy she loses nothing if the baby is a girl she loses Downton Abbey and loses everything To Mathew he will be happy if Marry marries him and sad if not Being the next Earl can only make him a bit happier a The simple case The simple case The game consists of 2 stages At first stage Matthew decide whether to propose he can choose not to do so in order to avoid getting hurt if Marry turns him down At second stage Marry decides whether to say she loves Matthew If Mathew proposes and Marry says she loves Matthew they both get 2 if Mathew proposes and Marry says she doesn t love Matthew they both get 1 if Matthew doesn t propose but Marry says she loves Matthew they both get 1 if Matthew doesn t propose and Marry says she doesn t loves Matthew they both get 1 for avoiding mismatching Please describe the game in normal form and find all pure strategy Nash Equilibriums Then describe the game in extended form the game tree and find the SPNE b Incomplete information Incomplete information The baby is born and Matthew knows the baby s gender but Marry doesn t for she is currently travelling to Roma Matthew writes to propose to Marry but doesn t tell her the baby s gender If the baby is a boy and Marry accepts the proposal Matthew gets 4 and Marry gets 0 If the baby is a boy and Marry turns Matthew down Matthew gets 1 and Marry gets 3 If the baby is a girl and Marry accepts the proposal Matthew gets 6 and Marry gets 4 If the baby is a girl and Marry turns Matthew down Matthew gets 3 and Marry gets 1 The probability that the baby is a boy is 0 5 Will Marry accept the proposal Solution to PS5 1 1 Wall at Silicon ValleyWall at Silicon Valley a Cournot solution F O C b If collusion F O C c First Gates will decide his production given Jobs decision Taking Gates reaction function into consideration Jobs choose production to maximize his profit thus Thus Gates will produce and while 2 2 Petroleum CompetitionPetroleum Competition a so their profit will be b China petro s reaction function is derived through solving Incorporate this into Sinopec s optimization problem we can get so Their respect profit is We can see that Sinopec s profit though increases is less than China petro Thus Sinopec doesn t have any advantages 3 3 Battle of the MarneBattle of the Marne First we can be sure that to France W is always better than F so we can delete F This leaves us a 3 2 matrix Now we suspect that we may delete Germany s R option for it never yields any better outcome than the other two We can test that by assign a positive probability to R Assume France s strategy is 0 p 1 p for we already delete France s option F so F is assigned with probability equal to 0 To Germany Obviously it will never happen In fact if we take a close look we can find that R will never produce any outcome than F if p 1 their expected outcome equals however contradicts our formal assumption Germany s R option is assigned a positive possibility Thus we can safely delete Germany s option R and face a familiar 2 2 matrix To Germany To France Thus the Nash Equilibrium will be 4 4 DilemmaDilemma at Downton Abbey at Downton Abbey a Normal form P stands for propose N stands for not to propose A stands for accept and R stands for reject We can find 3 pure strategy NE very quickly they are Extensive form From backward induction you can find the only SPNE is Matthew proposes to Marry and Marry accepts that b We can calculate the threshold belief that Marry will accept Matthew Let r stands for the threshold belief that the baby is a boy Since the probability that the baby is a boy is 0 5 and Matthew will propose whenever the baby is a boy or not the belief can t be updated and so Marry will accept Matthew 1 Intermediate Microeconomics PS6 Fall 2011 1 Muskrat Ontario has 1 000 people Citizens of Muskrat consume only one private good Labatt s ale There is one public good the town skating rink Although they may differ in other respects inhabitants have the same utility function This function is Ui Xi Y Xi 100 Y where Xi is the number of bottles of Labatt s consumed by citizen i and Y is the size of the town skating rink measured in square meters The price of Labatt s ale is 1 per bottle and the price of the skating rink is 10 per square meter Everyone who lives in Muskrat has an income of 1 000 per year a Write out the equation implied by the Samuelson conditions b Show that this equation uniquely determines the efficient rink size for Muskrat What is that size 2 Consider a consumer with preferences for consumption of a composite good Y and leisure N given by the following utility f