Problem Set 3

3.185Problem Set3

More di?usion

1.Decarburization of solid low-carbon steel(20)

A steel sheet1mm thick with0.05wt%carbon is held in air at1000?C for an hour to remove the

carbon.At that temperature,the di?usivity is10?6cm2

s .(It is this high because carbon is an interstitial

solute.)

(a)If the surface concentration of carbon is zero,about how long will it take for the carbon concen-

tration at the center of the sheet to start to fall?(Hint:consider the solution for a semi-in?nite medium.)(6)

(b)How long will it take for the maximum carbon concentration(in the center)to reach0.005wt%?

(8)

(c)What is the maximum carbon concentration(again in the center)at the nominal“time to reach

steady-state”(i.e.thickness2/D)?(6)

2.A thin metal foil is used to separate hydrogen from other gases with lower di?usivity in the metal.(12)

(a)The metal foil is0.2mm thick,and H2partial pressure is0.2atm on the hydrogen-rich side.If the

temperature is400?C and there is negligible hydrogen on the other side(since it’s being pumped away),calculate the?ux of hydrogen through the foil in cubic centimeters at STP per unit area per second for nickel,copper and aluminum.(Use the table on p.466of Poirier and Geiger.)(6)

(b)At what hydrogen partial presure(on the H2-rich side)will the?ux be half of what you calculated

in part2a?Brie?y explain the?ux/pressure scaling in terms of thermodynamics and/or kinetics.

(6)

3.Growth of a TiSi intermetallic layer(20)

Titanium vapor is deposited on silicon at a high enough temperature to produce a high-quality smooth titanium?lm with few defects.At that high temperature,the titanium and silicon react,producing an intermetallic titanium-silicon compound at the interface,which acts as a di?usion barrier preventing aluminum interconnect wires deposited later from di?using into the silicon and disappearing.(A low-di?usivity AlTi layer also forms at the aluminum-titanium interface,but we’ll focus on TiSi here.) Suppose that layer is TiSi,that is,50mol%titanium,50mol%silicon.It is a contiguous layer,with uniform thickness,and its growth is limited by one of two mechanisms:

?Di?usion of titanium through the TiSi layer with di?usivity D.

?Reaction of silicon at Si-TiSi interface with reaction rate coe?cient k.

You may assume the following:

?The TiSi-Ti interface is at equilibrium.

?The concentration pro?le across the intermetallic layer is linear(?.e.pseudo-steady-state di?u-sion).

?Ignore changes in the thickness of the Ti?lm over time.(That is,for simplicity,assume the whole Ti?lm is deposited,then the intermetallic layer starts to grow.)

?The(simpli?ed)phase diagram looks something like:

(a)Write the mass transfer Biot number which compares the limiting reaction and di?usion mecha-

nisms.Which mechanism limits the growth rate if this number is large?(6)

(b)Sketch the concentration of titanium C Ti in mol%as a function of distance from the top surface

z for small and large values of the Biot number.Which corresponds to a“short time”,and which to a“long time”?(7)

(c)Sketch the relationship between TiSi layer thickness Y and time t,giving the exponents of both

short-and long-time behavior(i.e.the n in Y∝t n).(7)

4.Iron oxidation(13)

A thick iron slab is heated to1000?C to prepare it for rolling into plate form.It spends about30

minutes at that temperature,and assume it is cooled very quickly during rolling.

(a)Using the table on p.498of the Poirier&Geiger,calculate the approximate mass of iron oxides

per unit area before rolling.Assuming most of this is Fe2O3,calculate the approximate thickness of the oxide layer.(8)

(b)If1%titanium is added to the iron,how does this change your answer to part4a?(Compare weight

gain,not thickness,since you’d need the density of the resulting oxide to determine thickness.)

(5)

5.Dimensional analysis:catalytic combustion of carbon monoxide(35)

CuO is a good catalyst for the reaction

2CO(g)+O2(g)?2CO2(g)

CuO is coated on the internal surfaces of porous Al2O3spheres of radius R,so the gases containing CO and O2can di?use in through the pores with di?usivity D and react on the internal CuO surfaces.

(The Al2O3is used as a substrate because of its high strength and stability at the high temperatures reached due to combustion;porous CuO would sinter into dense solid spheres at such temperatures.) Because the pores are a lot smaller than the sphere,you can treat this as a homogeneous chemical reaction throughout the porous solid.Assume for this problem that O2is the limiting reagent,and

the rate of reaction is proportional to its concentration with rate constant k,so G=?kC O

2.The

concentration of oxygen in the pores just inside the surface of the sphere is approximately equal to that in the gas outside,so that is our outer surface boundary condition.

If the concentration of oxygen throughout each sphere is very uniform,then the reaction will be occurring on all of the inner pore surfaces within each sphere,so the CuO will be well-utilized.On the other hand,if we are?owing CO gas through this bunch of spheres,the larger the spheres,the less resistance to?ow there will be.So we want the spheres to be as large as possible without being so large that there is no reaction at their center.

(a)Write an expression for the dimensionless parameter which describes the uniformity of oxygen

concentration,in terms of the other problem parameters.(5)

(b)Identify the number of parameters,and the number of independent units,and use the Buckingham

pi theorem to determine the number of independent dimensionless parameters in the problem.(10) (c)Write a new expression for the uniformity parameter in part5a in terms of the other dimensionless

parameter(s).(5)

(d)If you double the sphere radius,how much does k have to increase to yield the same unifor-

mity?(The reaction rate coe?cient k tends to be a much stronger function of temperature than di?usivity,so changing the temperature will change D a small amount and k a large amount.)(7) (e)Draw a sketch(or sketches)showing how you think that uniformity parameter will vary as a

function of the other parameter(s),making sure your sketch captures the transition from strongly non-uniform to roughly uniform concentration in a sphere.(8)

Practice Problem Set-1

Practice Problem Set – 1 ( The following problems are from Corporate Finance by Ross, Westerfield, and Jaffe – Tenth edition, McGraw-Hill / Irwin – ISBN 978-0-07-803477-0 ) 1.Audrey Sanborn has just arranged to purchase a $ 550,000 vacation home in the Bahamas with a 20 percent down payment. The mortgage has a 6.1 percent stated annual interest rate, compounded monthly, and calls for equal monthly payments over the next 30 years. Her first payment will be due one month from now. However, the mortgage has an eight-year balloon payment, meaning that the balance of the loan must be paid off at the end of year 8. There were no other transaction costs or finance charges. How much will Audrey’s balloon payment be in eight years ? 2.Your job pays you only once a year for all the work you did over the previous 12 months. Today, December 31, you just received your salary of $ 65,000, and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 5 percent of your annual salary in an account that will earn 10 percent per year. Your salary will increase at 4 percent per year throughout your career. How much money will you have on the date of your retirement 40 years from today ? 3. A 15-year annuity pays $ 1,500 per month, and payments are made at the end of each month. If the interest rate is 12 percent compounded monthly for the first seven years, and 6 percent compounded monthly thereafter, what is the present value of the annuity ? 4.You are saving for the college education of your two children. They are two years apart in age; one will begin college 15 years from today and the other will begin 17 years from today. You estimate your children’s college expenses to be $ 45,000 per year per child, payable at the beginning of each school year. The annual interest rate is 7.5 percent. How much money must you deposit in an account each year to fund your children’s education ? Your deposits begin one year from today. You will make your last deposit when your oldest child enters college. Assume four years of college. 5. An insurance company is offering a new policy to its customers. Typically the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: the purchaser ( say, the parent ) makes the following six payments to the insurance company: Birthday First Second Third Fourth Fifth Sixth Deposit ($) 500 600 700 800 900 1,000 After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $ 275,000. If the relevant interest rate is 11 percent for the first six years and 7 percent for all subsequent years, is the policy worth buying ?

problem_set_2

7.36/7.91/BE.490 家庭作业 2 上交时间：3月11号下午 1:00 注意：请查阅课程网站获取如何提交程序设计方面问题的电子版。另外，在本次作业结尾有一些关于编程方面的有用的提示。 1．古基因组学 I ——BLAST检索核酸序列 你的实验室已经找到一种从恐龙化石中提取出DNA序列的新方法，现在为了研究恐龙的进化史和鸟类的起源问题，你们从翼龙的化石中提取出了DNA序列。 该序列保存在dino1.fa文件中。 a. 在NR数据库中使用BLASTN进行检索，在使用缺省参数设置时所得到的检索结果 的第一条序列是什么？它看上去正确吗？为什么？ b. 将错配的罚分从缺省值（-3）改成-1，再检索一次，结果的第一条序列是什么？结 果中的E-value值是多少？跟a的结果相比检索质量上是否有不同？降低错配罚分 的值是否得到了你期望的结果？ c. 如何使用BLASTX检索？使用BLASTX检索结果的第一条序列是什么？结果中的 E-value值和bit得分是多少？它看上去是正确的吗？该生物属于哪一类？它能解释 翼龙的进化问题吗？ d. 试解释为什么BLASTN和BLASTX的结果差异很大。 2．古基因组学 II——BLAST用于基因内含子的影响 你得到了一条新的翼龙的序列，保存在dino2.fa中，你尝试使用BLASTN和BLASTX在NR数据库及蛋白质数据库中检索，但未得到相关结果（请尝试一次确认一下）。由于该序列来自翼龙基因组的富含基因区，因此你怀疑也许是序列中包含了一段未知基因使得BLAST无法正确匹配，为了验证该可能性，你决定对翼龙序列的模体结合位点进行研究，看是否能够提取出dino2.fa序列中的基因外显子。在进行了多次检索后，你找到了56个翼龙基因组DNA片段的BLASTX结果及对应的已知蛋白质序列，同时推断这些结果中很可能包含外显子和结合位点，由于BLASTX的结果经常无法准确的对应外显子的边界，（例如，因为在靠近结合区氨基酸的改变造成序列的删节、在结合位点/内含子的转录错误导致序列延长）你建立了一个数据库，大约包含BLASTX检索结果的从前往后数的前25个碱基，命名为“3primesplicesites.txt”（为的是找到3’的结合位点模体），另一数据集包括BLASTX检索结果的从后往前数的25个碱基，命名为“5primesplicesites.txt”（为的是找到5’的结合位点模体）。 a. 使用以上2个数据集运行MEME（https://www.wendangku.net/doc/3d1595727.html,/meme/website/meme.html）， 找到56个序列中的共同模体作为恐龙3’和5’的结合位点，你找到了哪些模体？使 用如下所示的单字符编码列出相同的模体： 注意：当使用MEME进行“多级相似序列”输出时将会出现问题，第四条核苷序列不会在模体的指定位置输出。因此，检查“Simplified pos.-specific probability matrix”或者“Information content”图表以确定模体的任意位置都包含所有的4个核苷。 b. 编写python程序读入两条相似序列，找到在“dino2.fa”序列中推断的内含子，并输 出预测结合的mRNA。程序应从命令行接收2个参数，即文件名，其一为包含2条 相似序列的文件，每条序列占一行（分别是3’和5’结合位点的相似序列），每条序

[耶鲁大学开放课程：博弈论].Problem.Set.2

耶鲁大学公开课：博弈论 习题集2（第4-5讲内容） Ben Polak, Econ 159a/MGT522a. 由人人影视博弈论制作组Darrencui翻译 1.回顾罚球的案例：裁判判罚给参与人1一次点球的机会，参与人1即将执行判罚。她有三种射门路径：左路、中路、右路。参与人2是门将。他可以选择防守左路、中路或者右路。两名参与人的行为同时发出。收益（以达成目概率的十倍计算）如下： 2 (a). 对于每一个参与人来说，有哪个策略严格劣于另一个（纯）策略吗？ (b). 在对参与人1的策略存在何种信念下，参与人2会觉得策略m是最佳对策？在对参与人2的策略存在何种信念下，参与人1会觉得策略M是最佳对策？[提示：本题不需要绘制三维图像！] (c). 假设参与人2站在参与人1的立场上思考后发现，无论参与人1存在何种信念，她都会选择改信念下的最佳对策。在这种情况下，参与人2是否应该选择策略m呢？ (d). 这个博弈是否存在（纯策略）纳什均衡？ 2.回顾合伙人案例（Watson书中习题）：回顾一下我们在第四讲中提到的商业合伙人的案例。两名律师合伙开了一家律师事务所并且平分收益。每名律师都要各自打算一下自己要为事务所付出多少劳动。事务所的收入按照如下公式计算：，其中、分别表示律师1和律师2付出的劳动量。参数反映了两人的协同效果：一名律师付出越多的辛劳，合伙人就会获得越多的收益。假设并且。两名律师的收益分别是： 其中表示劳动的成本（注意：边际成本递增）。假设这家律师事务所没有其它的开销。在课堂上我们论证了，理性策略（即迭代剔除非最佳对策后剩余的策略）是

(a). 假设两名律师达成一致，决定两个人都付出一样多的劳动，并通过合同的形式规定了劳动量的指标。如果他们想要最大化净收益（即收益减去劳动成本），他们应该在合同中规定各自付出多大的劳动量呢？这与课堂上得出的理性策略的劳动量相比有什么不同？[提示：为了解题方便，可以暂时考虑b=0的特殊情况] (b). 假设第(a)题中的合同只对合伙人2有约束力，即合伙人2需要按照要求中的付出等量的劳动，而合伙人1可以任意在[0,4]的劳动量中自由选择。合伙人1会选择付出多少劳动呢？这与 和有什么不同吗？请给出简明的解释。 (c). 回到最开始的博弈状态，假设现在，即合伙人的辛勤劳动起到了反协同效果。求出这种情况下的最佳对策函数，并绘制相应的函数图像，找出这种情况下对应的理性策略。把它与(a)中的指标作比较。[提示：并不需要重做(a)的全部过程] 3. 纳什均衡与迭代剔除（Gibbons教科书上的习题）：请看下面的这个博弈： (a). 哪些策略不会被迭代剔除严格劣势策略的过程剔除？ (b). 找出此博弈的（纯策略）纳什均衡 (c). 请尽可能详尽地解释说明，通常情况下（并不要局限于此博弈），组成纳什均衡的策略是否无法被迭代剔除严格劣势策略的过程剔除？ 4. 分钱计划（Gibbons教科书中习题）：参与人1和参与人2因为如何分配10美元的问题争执不休。每个参与人都说出了一个自己预期金额，该金额在0到10之间且允许出现小数。两人需要同时做出选择。参与人的收益就是她分得的钱款。这个博弈有两条规则。无论按哪条规则来分钱，如果出现的情况，每人获得自己的预期金额，剩余的钱款被销毁。 (a).第一条规则是，如果，那么每个参与人都一无所获并且钱会被销毁。这种情况下的（纯策略）纳什均衡是什么？ (b).第二条规则是，如果，并且每个人的预期金额是不同的，那么预期金额最小的参与人分得等值的钱款而剩余的钱款归另一个参与人。如果，并且，那么每个人都分得5美元。这种情况下的（纯策略）纳什均衡是什么？ (c).假如我们为前两条规则增加一个限制条件，即预期金额必须是整数。这是否会改变前两条规则下的（纯策略）纳什均衡？

Hk5120ProblemSet2-14L4_r1

The Hong Kong University of Science & Technology Department of Finance FINA 5120: Corporate Finance L4 Fall 2014 Dr. Ahron Rosenfeld Problem Set #2 Guidelines for the analysis: a.Please submit your answers electronically through LMES on or before 7pm on September 25, 2014. b.The submission file must be printer‐friendly. c.The use of Excel functions is strongly encourage d. d.The grade for this homework weighs 5% of the final grade in the cours e. e.The analysis should contain answers to the following questions: Question 1: The XYZ Corporation is considering the installation of a new conveyer system to replace the existing one. The new system has a purchase price of $15,000, useful life of 5 years, and no terminal salvage value. It would reduce annual labor costs by $5,000 per year, require the level of inventory to be higher by $2,000 but would leave all other annual cash flows (except taxes) unchanged. The existing system was purchased 3 years ago for $10,000 with an estimated life of 8 years for tax purposes and no salvage value. It could be sold now for $3,000 or used for the remaining 5 years and then sold for $2,000. Assume the firm uses straight line depreciation on all equipment. The tax rate on income is 40 percent. The appropriate discount rate is 10 percent. a.Show the change in the firm’s free cash flows for each of the next 5 years for the new system. Specify the timing of each item in your answer. b.Show the change in the firm’s free cash flows for each of the next 5 years for the old system. Specify the timing of each item in your answer. c.Would you recommend replacing the system? Why? Question 2: a.In the Ocean Carriers case, use the DCF method to determine if Ms. Linn should purchase the $39 million capesize? Assume that Ocean Carriers is a U.S. firm subject to 35% taxation. Ocean Carriers uses a 9% annual discount rate. b.What do you think of the company's policy of not operating ships over 15 years old? Agree/disagree, why?

problem set 2

1.3. Describe how, if at all, each of the following developments affects the breakeven and actual investment lines in our basic diagram for the Solow model: (a) The rate of depreciation falls. (b) The rate of technological progress rises. (c) The production function is Cobb–Douglas, f (k) = kα,and capital’s share,α, rises. (d) Workers exert more effort, so that output per unit of effective labor for a given value of capital per unit of effective labor is higher than before. 1.4. Consider an economy with technological progress but without population growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of workers. (a) At the time of the jump, does output per unit of effective labor rise, fall, or stay the same? Why? (b) After the initial change (if any) in output per unit of effective labor when the new workers appear, is there any further change in output per unit of effective labor? If so, does it rise or fall? Why? (c) Once the economy has again reached a balanced growth path, is output per unit of effective labor higher, lower, or the same as it was before the new workers appeared? Why? 1.5. Suppose that the production function is Cobb–Douglas.

Problem_Set_3_solutions

AP/ECON3150International Trade Problem set3Solutions Problem1 a.Answer:the problem says that Home is K-abundant and that steel production in K-intensive.Therefore,Home’s PPF will be skewed towards output. steel output while Foreign PPF will be skewed towards wheat b.Answer:Home no-trade equilibrium is at point E A,and Foreign no-trade equilibrium is at point E A.We will use subindex A to denote‘autarky’and T to denote‘trade’.With identical utilities,Home’s no-trade production and consumption of S and W corresponds to a relative price of wheat that is higher than that in Foreign.This is shown by the steeper-sloped relative price line in Home, p W p S A,than in Foreign, p W p S A.Di¤erences in capital and labor stocks between these two countries lead to Home’s comparative advantage in steel and p S p W A< p S p W A. 1

MIT麻省理工学院 算法导论公开课Problem Set 1

Introduction to Algorithms September7,2005 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik D.Demaine and Charles E.Leiserson Handout5

(a) (b) (c) (d)and(note the little-) (e)and Problem1-2.Recurrences Give asymptotic upper and lower bounds for in each of the following recurrences.Assume that is constant for.Make your bounds as tight as possible,and justify your answers. (a) (b) (c) (d)

Figure1:An example of a convex polygon represented by the array.is the vertex with the minimum-coordinate,and are ordered counterclockwise. (b)Give an algorithm to?nd the vertex with the maximum coordinate in time. (c)Give an algorithm to?nd the vertex with the maximum coordinate in time.

北大光华微观经济学作业 Problem Set-1

Problem Set 1 1. Draw the indifference curves for the following individuals’ preferences for two goods: hamburger and beer. a) Al likes beer but hates hamburger. He always prefers more beer no matter how many hamburgers he has. b) Chris eats one hamburger and washes it down with one beer. He will not consume an additional unit of one item without an additional unit of the other. c) Doreen loves beer but is allergic to beef. Every time she eats a hamburger she breaks out in hives. 2. 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? 4．Assume that a utility function is given by Min{X, Y}, where Min is the minimum of the two values of X and Y. What is the Slutsky equation that decomposes the change in the demand for X in response to a change in its price? What is the income effect? What is the substitution effect? 5．Using Slutsky equation shows that Giffen goods must be inferior goods, but inferior goods need not be Giffen goods. 6．Consider a consumer who purchases two goods, x and . The consumer’s utility function is . The prices are given by y 2(,)U x y xy =x P and , and the consumer’s income is y P I . 1) Set up the Lagrangian expression for solving the consumer’s utility maximization problem (subject to budget constraint) and write down the first order conditions. 2) Derive the demand functions for goods x and y .

Advanced Econometric - Problem Set 1高级计量经济学练习题

Advanced Econometrics-Problem Set I (Due on Sep.19,2018in class) 1.Suppose X and Y have the joint density given by f(x,y)=8xy for0≤x≤y≤1. (a)Compute E[X],E[Y],and E[X/Y]. (b)Notice that E[X/Y]=E[X]/E[Y].Is it true in general?If not,explain brie?y why this holds here. 2.(a)Suppose Y～N(0,1).Prove E[Y k]=0for k=1,3,5,7,.... (b)Suppose E[Y|X]=0.Show that Cov(Y,X)=0. (c)Does zero correlation imply independence? 3.Suppose{Y i}n i=1 is a sequence of i.i.d.observations with common meanμY and common varianceσ2Y.Consider the linear estimator?μ=∑n i=1a i Y i. (a)Provide a condition on{a i}n i=1such that?μis an unbiased estimator ofμY. (b)Calculate the variance of?μ. (c)Show thatˉY is the best linear unbiased estimator(BLUE). 4.Let X be a T×k matrix whose columns are linearly independent,and let M≡ I T?X(X X)?1X . (a)Show that M is symmetric and idempotent,and calculate the eigenvalues and rank of M.Is M positive semi-de?nite? (b)In the special case where X is a T×1vector of ones,what form does My take,where y=[y1,...,y T] . 5.Let P be a nonsingular symmetric(k1×k1)matrix,Q a nonsingular symmetric (k2×k2)matrix.Verify the following formula for the inverse of a partitioned matrix: P R R Q ?1 = W?WRQ?1 ?Q?1R W Q?1+Q?1R WRQ?1 , where W=(P?RQ?1R )?1. 1

problemset4

Problem Set4. Ben Polak Econ159a/MGT522a Three questions due October10,2007 (1)Comparative Statics of Mixed Strategy Equilibria.Consider the following two-player game. l r U12;23;9 D5;84;2 (a)Find all the Nash equilibria,pure and mixed.Explain how you know you have found all the equilibria. (b)Suppose that the payo of the column player u2(D;l)is reduced from8to6,but all other payo s remain the same.Again, nd all the pure-and mixed-strategy Nash equilibria. (c)Compare the mixed-strategy equilibria in parts(a)and(b).Did this worsening in one of player 2's payo s change player2's equilibrium mixed strategy?Did it change player1's?Give some intuition. 2.On Her Majesty's Secret Service.The famous British spy001has to choose one of four routes a,b,c,or d(listed in order of speed in good conditions)to ski down a mountain.Fast routes are more succeptible to avalanche.At the same time,the notorious rival spy002has to choose whether to use(\y")or not to use(\x")his valuable explosive to try to cause an avalanche.The payo s to this game are as follows. 002 x y a12;00;6 001b11;11;5 c10;24;2 d9;36;0 (a)Let p1(x)be the probability assigned by001's belief to002's playing x.Explain what001 should do if p1(x)>2 3;if p1(x)<2 3 ;and if p1(x)=2 3 ? (b)Suppose you are Mr.Queue,the Yale-educated technical advisor to British military intel- ligence.Are there any routes you would advise001certainly not to take?Explain your answer.

problem set 1

Political Economy of Institutions and Development:14.773 Problem Set1 Due date:February,29,2008. Question1 1.Consider the example of a three-person three-policy society with pref- erences 1a b c 2b c a 3c b a Voting is dynamic:…rst,there is a vote between a and b.Then,the winner goes against c,and the winner of this contest is the social choice.Find the subgame perfect Nash equilibrium voting strategy pro…les in this two-stage game in“weakly undominated”(recall that each player’s strategy has to specify how they will vote in the…rst round,and how they will vote in the second round as a function of the outcome the…rst round).[Hint:for“weakly undominated”strategies,…rst eliminate weakly dominated strategies in the last round,and then eliminate whatever is weakly dominated in the previous round,etc.]. 2.Suppose a generalization whereby there are…nite number of policies, Q=f q1;q2;:::;q N g and M agents(which you can take to be an odd number for simplicity).Voting takes N 1stages.In the…rst stage, there is a vote between q1and q2.In the second stage,there is a vote between the winner of the…rst stage and q3,until we have a…nal vote against q N.The winner of the…nal vote is the policy choice of the society.Prove that if preferences of all agents are single peaked(with a unique bliss point for each),then the unique subgame perfect Nash equilibrium in“weakly undominated”implements the bliss point of the median voter. 1

Problem Set 1 DuDu

Problem Set 1 of FINA 3404 Instructor: Dr. Du Du Select the best answers for questions (2 points each for questions 1-8) 1.Of the following exchange rates, which are quoted in European terms? (Answer: D) i) 2 USD for one British pound; ii)0.75 euro for one USD; iii)7.8 HKD for one USD; iv)100 yen for one USD A)only i) B)only ii) C) iii) and iv) D)ii), iii) and iv) 2.Which of the following statements about forward rate/forward contract is FALSE? (Answer: D) A)the forward rate is determined when both the long side and the short side enter into the forward contract B)The forward rate will not be executed until the forward contract expires C)The forward rate represents the market prediction of the future spot rate D)The forward rate is determined when the forward contract is about to expire 3. A WSJ (wall street journal) article reports that: “the euro was at $1.3653 up from $1.3674 at Monday’s close, while the dollar was at 115.09 yen from 116.08 yen” (answer: A) That means the dollar_________against euro, and the yen_____________against dollar A)appreciated; appreciated B)appreciated; depreciated C)depreciated; depreciated D)depreciated; appreciated 4.The AUD/$ spot exchange rate is AUD1.60/$ and the SF/$ exchange rate is SF1.25/$.