2011春中微2班习题

Intermediate Microeconomics Spring 2011 Hand in Form Hand Writing Only Assigned Time 2011 3 5 Due Time by 2011 3 10 7 40pm 1 Vivi wants to exercise and has a choice of going to dance classes with a cost 3 each time or doing yoga with a cost of 2 per visit She likes to do some of each type of exercise that is the more dance she does the more she enjoys yoga and vice versa However she enjoys dancing even if she never does yoga while yoga is fun for her only if she also does some dances The following utility function reflects Vivi s preference 4uxyx where x is the number of dance classes per semester and y is the number of visits to the yoga class Think about how this utility function represents Vivi s preference as described Vivi s budget for exercising is 100 per semester We want to determine the number of dance classes and visits to the yoga class that she will choose Or stated another way we want to determine the number of dance and yoga classes that will make her most happy given her budget of 100 Hint a Give the formula for Vivi s budget constraint Express the constraint as y being a function of x b Give the formula for Vivi s MRS of x for y c Suppose Vivi spends her entire budget on exercise solve for her optimal x and y 2 Mike s utility for swim x and gym y is 254uxyxy Swim costs 10 each time gym costs 8 per visit and his budget is 400 a Describe how Mike s preference over swim and basketball as reflected in this utility function differs from Vivi s as described in the previous question b Determine Mike s optimal choice in spending his money on swim and gym 3 Elena got I from her father as birthday present and she decided to spend the money either taking a piano class or a golf class The piano class charges 1 P each time and golf class charges 2 P each time Her utility function is 0 50 5 1212 2U X XXX a Suppose I 800 1 P 25 2 P 16 derive her individual demand curve in terms of I 1 P 2 P and find her optimal decision and maximum utility b Suppose U 40 1 P 25 2 P 16 derive her compensated demand curve in terms of U 1 P 2 P and find her optimal decision and minimum expenditure c Compare the results in a and b describe their relationship using your own words 4 Suppose that good A is an inferior good and good B is a normal good and the consumer spends all of his money on these two goods a If the price of good A rises what will happen to the consumer s demand for Good A Explain briefly in terms of income and substitution effects b If the price of good B rises what will happen to the consumer s demand for good A Explain briefly in terms of income and substitution effects c Are these goods complements or substitutes Explain 1 Vivi wants to exercise and has a choice of going to dance classes with a cost 3 each time or doing yoga with a cost of 2 per visit She likes to do some of each type of exercise that is the more dance she does the more she enjoys yoga and vice versa However she enjoys dancing even if she never does yoga while yoga is fun for her only if she also does some dances The following utility function reflects Vivi s preference 4uxyx where x is the number of dance classes per semester and y is the number of visits to the yoga class Think about how this utility function represents Vivi s preference as described Vivi s budget for exercising is 100 per semester We want to determine the number of dance classes and visits to the yoga class that she will choose Or stated another way we want to determine the number of dance and yoga classes that will make her most happy given her budget of 100 Hint a Give the formula for Vivi s budget constraint Express the constraint as y being a function of x b Give the formula for Vivi s MRS of x for y c Suppose Vivi spends her entire budget on exercise solve for her optimal x and y a The budget constraint is 32100 xy then as required we can transform this constraint into 3 50 2 yx b On a given IC we have u x yconst Then we have 0 uu dxdy xy i e 4 u dyy x u dxx y c max4 32100 410032 430 20 100320 43 2 100320 18 23 xyxstxy Lxyxxy L y x L x y xy y x xy x y 2 Mike s utility for swim x and gym y is 254uxyxy Swim costs 10 each time gym costs 8 per visit and his budget is 400 a Describe how Mike s preference over swim and basketball as reflected in this utility function differs from Vivi s as described in the previous question b Determine Mike s optimal choice in spending his money on swim and gym b max 254 108400 254400 108 25 100 2480 400 1080 255 244 400 1080 20 25 xyxystxy Lxyxyxy L y x L x y xy y x xy x y 3 Elena got I from her father as birthday present and she decided to spend the money either taking a piano class or a golf class The piano class charges 1 P each time and golf class charges 2 P each time Her utility function is 0 50 5 1212 2U X XXX a Suppose I 800 1 P 25 2 P 16 derive her individual demand curve in terms of I 1 P 2 P and find her optimal decision and maximum utility b Suppose U 40 1 P 25 2 P 16 derive her compensated demand curve in terms of U 1 P 2 P and find her optimal decision and minimum expenditure c Compare the results in a and b describe their relationship using your own words a 0 50 5 121122 max 2 XXst PXP XI 12 12 22 II XX PP Thus the demand curve is 12 12 22 II PP XX 12 16 25 40 XX U b 0 50 5 112212 min 2PXP XstXXU 21 12 12 22 hh PPuu XX PP Thus the demand curve is 22 21 1222 12 22 hh U PU P PP XX 12 16 25 800 hh XX Exp c Utility maximum problem and expenditure minimum problem is equivalent 4 Suppose that good A is an inferior good and good B is a normal good and the consumer spends all of his money on these two goods a If the price of good A rises what will happen to the consumer s demand for Good A Explain briefly in terms of income and substitution effects b If the price of good B rises what will happen to the consumer s demand for good A Explain briefly in terms of income and substitution effects c Are these goods complements or substitutes Explain a h AAA A AA xxx x ppI 0 h A A x p Good A is inferior so we have0 A x I Thus if the price of good A rises whether its demand will increase of decrease is uncertain b h BBB B BB xxx x ppI 0 h B B x p Good B is normal so we have0 B x I Thus if the price of good B rises its demand will certainly decrease c h AAA B BB xxx x ppI 0 A x I Good A is inferior Additionally by hh AABB p xp xe p u we have0 hh AB AB BB xx pp pp Thus we have 0 h A B x p and0 A B x p Therefore good A is good B s net substitute and gross substitute Similarly it is easy to find that good B is good A s net substitute but not necessary gross substitute This case tells us that net substitute is symmetric but gross substitute is asymmetric Intermediate Microeconomics Spring 2011 请手写作业 请于3月19日15点10分前上交 1 假设北大学生京霜的效用函数为 x 为在食堂消费的食物 y 为在其他物品上的消 费量 y 的价格标准化为 1 该学生的收入为 3 6 x 的价格开始为 0 36 随着物价的上涨 食堂将 x 的价格上调为 1 1 请计算当 x 的价格为 0 36 时的 x 的消费量 y 的消费量 当 x 的价格上涨为 1 时 x 的消 费量 y 的消费量 2 请计算食堂价格上涨所引起的 x 消费量的变化 其中收入效应以及替代效应各是多少 3 请计算食堂价格上涨所以其的 y 消费量的变化 其中收入效应以及替代效应各是多少 4 图形表示 2 3 中的变化 请标出收入效应部分与替代效应部分 5 京霜的父亲想保证孩子的效用和涨价前相同 那他应该多给孩子多少钱呢 京霜的母亲 却觉得应该能保证孩子在涨价后仍然可以买得起涨价前的消费量 那她应该多给孩子多 少钱呢 京霜会更喜欢他父亲的做法还是母亲的做法呢 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 假设你的效用函数为 u x1 x2 x3 为类似商品 价格关系为 1 1 x1 x2 x3 可以看作组合商品吗 如果政府决定对 x1 x2 x3 征税 使其价格都上 升 若政府增加税负使 增大 这时还能把这三种商品看成组合商品吗 2 若 0 则原效用函数变为什么形式 在新效用函数下 将 x1 x2 x3 看作组合商品 z 分别求 z 和 x4 的补偿需求函数 4 请你判断一下 并解释 1 A 同学收入都来自爸妈给的固定零花钱 他天天宅在宿舍 钱全部用来买泡面和香 肠 一次 他说 泡面越贵我买的越多 又一次 他说 这个月要是能拿到奖学 金的话 我发誓少买点香肠了 据你观察 近期泡面和香肠价格没有变化 那么你 觉得 A 同学说话靠谱 说话前后自洽 吗 为什么 提示 请将商品分类和需求 弹性相关的某些公式联系起来 2 如果需求曲线是一条直线 则直线上各点的需求价格弹性是一样的 这种说法正确 吗 为什么 Answers to PS2 1 Max u s t Px x Py y Applying Langrangian method we can get x y Thus V E 2u Then u u 1 0 36 5 1 8 1 5 1 8 2 3 3 2 2 1 2 3 0 1 2 1 2 4 5 So 5 1 1 8 1 6 8 His mother s proposal is preferred 2 At first all the income is spent on x According to the form of his utility function we can infer that Now the price of x changes from to X Y a all the income is still spent on x 0 b 0 c 3 1 x1 x2 x3 can be treated as a composite comodity After the tax they cannot be treated as a composite commodity Y X X Y X Y IC IC since any change in the commodity tax will change the relative price of the three 2 Now the utility function is u lnx1 lnx2 lnx3 lnx4 If we treat the x1 x2 x3 as a composite commodity z the price of z is pz p2 p3 and z Solve the expenditure minimization problem using Lagrangian method we can get the X1 x2 x3 x4 V ln E 4 Z 3 4 True or False 1 The A s words are not kaopu The more expensive the paomian is the more he will buy Then paomian is giffen good for him The more income he has the less xiangchang he will buy Then xiangchang is inferior good for him So paomian and xiangchang are both inferior goods for him Further according to the Cournut Aggregation in the two good world the two goods cannot be both inferior goods and are both negative Now we know the A s words are not kaopu 2 Obviously wrong 1 一个工人操作一台机器 每天生产 20 双鞋 因此生产函数为 Q 20Min L K 现在 厂主 有 K 台机器 每双鞋价格为 P 工人工资为 W 1 当 LK 时 工人的边际产值是多少 2 当 W 20P 时 该厂生产多少双鞋 当 W20P 意味着有亏损 因而不生产 maxL P 20pmin L k WL If L k 20PL WL 20P W L K 20PK WL 0 所以有违约动机 想多增加产量 3 此题结果同古诺竞争 古诺均衡产量为 q1 q2 1000 p 10 两个厂商各自获得 10000 的利 润 此题目表述有一点问题 如果对这题有不同理解 可暂时搁置争议 忘记这道题就可 以了 4 1 两个小贩都将位置选在公路中点位置为该博弈唯一的纯策略纳什均衡 证明此为唯一纳 什均衡的方法 首先证明这是一个 NE 即给定对方的位置在中点 我都不会改变我现在处 在中点的选择 然后证明其他的位置选择都不可能满足 NE 的要求 可分为两种情况去证明 一是两个小贩的位置不重合 二是两个小贩的位置重合但不在中点 易知上述两种情况下 两小贩给定对方的位置 总是有动机去移动自己的位置选择 不满足 NE 的要求 2 证明方法类似第一问中的方法 分为三种位置排列情况去说明 一是三个小贩都不在一 个位置 二是两个小贩在同一位置 另外一个小贩在另外的位置 三是三个小贩都在同一个 位置 易知上述三种情况都不满足 NE 的要求 即可知该博弈中没有纯策略纳什均衡 5 6 对于任意的一个企业 它在不同条件下的产量 价格和利润分别为 c 为边际成本 c 0 为了使垄断价格可以作为子博弈完美纳什均衡的结果出现 那么合谋时企业利润的现值不应 低于背叛时的现值 从而解得贴现因子的最小值为 将其对 n 求导 可知 min与 n 的关系 7 从分析 B 的情况开始 记推测两种需求函数的情况分别为 1 和 2 B1 110 QA QB QB B2 70 QA QB QB 所以 110 QA 2 1 70 QA 2 2 再分析 A 的情况 Max A 0 5 90 QA QA 0 5 90 QA QA 所以 90 0 5 2 3 同时解 1 2 3 得 30 40 20 Problem set 6 1 Yidong Coal Company is the only hirer of coal workers in Erdos and thus it acts as a monopsonist The supply curve for coal workers is given by L 20w where L is the number of workers hired and w is their hourly wage Yidong Company s labor demand marginal revenue product curve is given by L 1200 10MRPl a How many workers will Yidong Company hire to maximize its profits and what wage will it pay b Assume now that the government implements a minimum wage law covering all coal workers If the minimum wage is set at 50 per hour derive the marginal expense curve faced by Yidong Company How many workers will Yidong Company now hire and how much unemployment will there be c Graph your results d How does a minimum wage imposed under monopsony differ in results as compared with a minimum wage imposed under perfect competition assuming the minimum wage is above the market determined wage 2 Bohai Sea is a highly productive fishing area off the east of China which can be divided into two zones in terms of fish population Zone 1 has the higher population per square mile but is subject to severe diminishing returns to fishing effort The daily fish catch in tons in Zone 1 is F x1 200 x1 2x1 2 where x1 is the number of boats fishing there Zone 2 has fewer fish per mile but is larger and diminishing returns are less of a problem Its daily fish catch is F x2 100 x2 x2 2 where x2 is the number of boats fishing in Zone 2 The fish are sold at 100 per ton The total cost capital and operating per boat is constant at 1000 per day a If any boats are allowed to fish where they want with no government restriction how many boats will fish in each zone What will be the gross value and total profit of the catch b If the government can restrict the number of boats in each zone how many boats should be allocated to each zone to maximize the net value of the fish catch What will be the gross value and total profit of the catch c There are 100 boats now licensed by the government to fish in these two zones If these boats are allowed to fish where they want with no government restriction how many boats will fish in each zone What will be the gross value and total profit of the catch 3 Suppose that two agents are deciding how fast to drive their cars Agent i chooses speed xiand gets utility ui xi from this choice we assume that ui xi 0 ui x i 0 be the cost that the accident imposes on agent i Assume that each agent s utility is linear in money a Show that each agent has an incentive to drive too fast from the social point of view b If agent i is fined an amount i t in the case of an accident how large should i t be to internalize the externality c If the optimal fines are being used what are the total costs including fines paid by the agents How does this compare to the total cost of the accident 4 A peculiar tribe of natives in the South Seas called the Grads consume only coconuts They use the coconuts for two purposes either they consume them for food or they burn them in a public religious sacrifice Suppose that each Grad i has an initial endowment of coconuts of wi 0 Let xi be the amount of coconuts that he consumes and let gi 0 be the amount of coconuts that he gives to the public offering The total number of coconuts contributed to