Econ 4721 Money and Banking Problem Set 2 Answer Key
|
|
- Martha Roberta Quinn
- 4 years ago
- Views:
Transcription
1 Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in ech period, ccording to N t nn t, where n.2. In ech period, young consumers re endowed with y 60 nd old consumers re endowed with 0 units of the single consumption good. Ech member of the genertions born in period nd lter hve the following utility function: u (c,t, c 2,t+ ) log c,t + β log c 2,t+ () with β 0.5. Members of the initil old genertion only live for one period nd hve utility u (c 0, ) log c 0,. The government expnds the money supply by fctor of z ech period, M t zm t. Assume tht z.5. The money creted ech period is used to finnce lump-sum subsidy of t+ goods to ech old person. () Solve for the (sttionry) Preto efficient lloction. The nswer should be two numbers ( ) c P O, c P 2 O. (b) Write the government s budget constrint in period t. (c) Define competitive equilibrium with money for this economy. (d) Solve for the rte of return of money (+ / ) nd the growth rte of the price level (p t+ /p t ) in sttionry equilibrium. The nswer should be two numbers. (e) Solve for the consumption lloction (c, c 2) nd lump-sum subsidy in sttionry equilibrium. The nswer should be three numbers. (f) Verify tht gents prefer the Preto efficient lloction to the competitive equilibrium lloction with infltion. (g) Illustrte the Preto efficient lloction ( ) c P O, c P 2 O nd the competitive equilibrium lloction (c, c 2) on the (c, c 2 ) plne. Your grph should lso include the fesibility line, the lifetime budget constrint, nd their indifference curves. Answer () (5 points) To find the Preto efficient lloction, we set up the following problem, explicitly incorporting the fct its solution will be sttionry: mx log c + β log c 2 (2) c,c 2 s.t. N t c + N t c 2 N t y (3)
2 Using the fct tht N t n N t, nd getting rid of N t terms, we rewrite constrint s: c + n c 2 y (4) You cn solve this problem in multiple wys. I use the quick solution formul I ve discussed in clss t some point: c P O c P O 2 So your nswer for prt () is ( ) c P O, c P 2 O (40, 24). (b) (5 points) The government budget constrint is very simple: y 40 (5) + β β 24 (6) + β y n (M t M t ) t N t (7) where the left hnd side (LHS) is the seignorge revenue, nd the RHS is the totl vlue of lump-sum rebtes tht re given to the people. (c) (5 points) We define the monetry equilibrium for this economy s follows: Definition. A competitive equilibrium with money for this economy would be n lloction ( ) c,t, c 2,t+ for every person born t time t, c 0, for the initil old, prices of money v t for ll t nd government trnsfers t for ll t such tht:. For every t, every person born t t, tking v t, v t+ nd t+ s given, chooses ( c,t, c 2,t+) to solve: 2. Initil old choose c 0, to solve, given v nd : mx log c,t + β log c 2,t+ (8) c,t,c 2,t+,m t c,t + vt m t y s.t. (9) c 2,t+ vt+m t + t+ mx c 0, log c 0, s.t. c 0, v m 0 + (0) 3. The government hs to pick t so tht it obeys its budget constrint for ll t : 4. All mrkets cler for ll t : v t+ (M t+ M t ) t+n t () N t c,t + N t c 2,t N t y (2) N t m t M t (3) where the first mrket-clering condition bove refers to the consumption mrket, nd the second refers to the money mrket. 2
3 (d) (5 points) To find the return rte on money, we use the sme trick s before. Strt with the money mrket clering condition: N t m t M t (4) nd multiply both prts by : N t ( m t ) M t (5) Now use the budget constrint for the first period of person s life to replce the expression in prentheses: N t (y c,t ) M t (6) nd you rrive t: N t (y c,t ) M t (7) Since t ws rbitrry, we cn write the sme expression for t +. Also, we focus on sttionry equilibrium, so c,t c for ll t, nd hence: We know tht /p t, so: + N t+(y c ) M t+ N t+ M t n N t(y c ) N t M t+ z M t 0.8 (8) + p t+ p t p t+ p t p t p t+ (9) (20) (e) (5 points) To solve for the sttionry CE lloction, you proceed s follows. First, you obtin the lifetime budget constrint, which will look s follows (I drop ll time subscripts since we tlk bout sttionrity here): c + c 2 y + (2) + + nd then you solve the individul gent s mximiztion problem. No mtter how you do it, you will rrive t something like this: c c 2 ( y + + β β + β ( vt+ v ) t + ) y + nd by (8) you cn simplify things s: c 2 ( ) (24) c 2 ( ) (25) (22) (23) 3
4 To find, we use wht we lredy know: tht c c nd we lso know tht in equilibrium, the consumption mrket hs to cler: N t c + N t c 2 N t y c + n c 2 y, so we cn plug ll tht we know into the eqution bove: ( ) 6 60, from where 6. And hence c 45 nd c 2 8. (f) (5 points) Verifying tht PO lloction is better thn CE lloction is very esy. In PO lloction, guy gets 40 when young nd 24 when old. So his totl lifetime utility is: u P O log 40 + log (26) 2 In CE lloction, guy gets 45 when young nd 8 when old. Hence his totl lifetime utility is: u CE log log < u P O (27) Though this my seem like smll difference, we only cre bout the fct tht utility is smller t the PO lloction. (g) (5 points) I expect you to drw digrm similr to this one: 4
5 Problem 2 (30 points) This problem mimics Exercise 3.4 from the CF book. Consider the following modifiction of our overlpping genertions model with consumers living for 2 periods. As usul, individuls re endowed with y units of consumption good when young nd with nothing when old, nd the good is not storble. The government keeps the fit money stock constnt, i.e. z. The popultion in the economy grows t rte n >. In every period, the government imposes lump-sum tx ech young person for τ units of consumption good. The totl proceeds of the tx re then distributed eqully mong the old popultion in this period. We ssume tht the subsidy to the old is less thn they would be hppy to consume when old (so tht there re still resons for people to hold fit money). Do the following: () Write down the first nd second period budget constrints tht regulr person born in period t fces. (Hint: remember tht in every period there re more young people live thn old people.) Derive the lifetime budget constrint. (b) Derive the rte of return on fit money in sttionry monetry equilibrium. (c) Is the sttionry monetry equilibrium lloction Preto efficient? Discuss. (d) Does this government policy hve ny effect on consumers welfre? Explin. 5
6 (e) Does your nswer to prt (d) of this question chnge if we ssume tht the subsidy to the old exceeds the quntity they would prefer to consume? (f) Assume now tht tx collection is costly, so for every unit of consumption collected from the young, only 0.5 units end up being vilble for distribution to the old? Does your nswer to prt (d) chnge? Comment. Answer () (5 points) It is importnt to relize tht the totl tx revenue in this economy will be N t τ, nd tht these must be distributed mong N t old people. Remembering tht N t nn t, we get tht ech old guy will get trnsfer of nd hence the budget constrints re: The lifetime constrint N t τ N t nτ, c,t + m t y τ c 2,t+ + m t + nτ m t (c 2,t+ nτ) + c,t + c 2,t+ y τ + nτ + + (b) (5 points) Demnd for rel money blnces in sttionry equilibrium is so the money mrket clering condition implies nd hence + m t y c τ, N t (y c τ) M t We cn now plug this into the lifetime budget constrint: (c) (5 points) The fesibility condition for the economy is N t+(y c τ) M t+ N t+ n. N t(y c τ) N t M t c + n c 2 y τ + n nτ c + n c 2 y N t c + N t c 2 N t y c + n c 2 y nd since it coincides with the lifetime budget constrint, it is cler tht monetry CE lloction will be Preto efficient. Since the money stock does not grow over time, this should not be surprising.. 6
7 (d) (5 points) The txtion-subsidy policy hs no welfre effects, becuse the tx nd subsidy cncels out from the lifetime budget constrint. So if the policy is cnceled, nothing would hppen. The only exception would occur if the tx ws lrge enough. (e) (5 points) If the tx is sufficiently lrge, people cnnot choose the optiml bundle they would choose in the bsence of the tx. For low vlues of τ, n individul cn freely choose the optiml bundle (if trnsfers do not bring up consumer s second-period income to c 2, he cn use money to get extr consumption). However, for lrger vlue of τ the individul cnnot hold the level of rel blnces to chieve the optiml point, becuse the trnsfer policy gives more to people when old thn they would ever hve chosen to purchse for themselves using money. (If we llowed people to hold negtive mounts of money, this problem would go wy. However, this won t work in equilibrium ll consumers re identicl nd jointly they cnnot borrow from nyone else except themselves.) The individul must settle for lower vlue of c nd becuse of this, the high vlue for the tx dversely ffects individul welfre. In this cse, ll second period consumption is finnced by the government trnsfer. (f) (5 points) In this cse, the second period subsidy will be 0.5nτ nd the lifetime constrint will chnge to c,t + c 2,t+ y τ + nτ, + + so in sttionry equilibrium c + n c 2 y τ 2. Thus the budget set would lie strictly within the fesible set. The monetry equilibrium could not chieve the Preto efficient lloction. In this cse, the tx/trnsfer system (due to its inefficiency) would hve negtive impct on welfre. It would be better to eliminte the tx/trnsfer system. Individuls cn provide for their own second-period consumption through fit money holdings. Problem 3 (35 points) There re two countries, USA (lbeled ) nd Chin (lbeled b). Ech country is described by our stndrd overlpping genertions model with consumers tht live for two periods. As usul, individuls in US re endowed with y units of consumption good when young nd with nothing when old, nd the good is not storble. (For Chin, the corresponding quntities re y b nd zero, respectively). Popultion in US grows ccording to the following lw of motion: Nt n Nt. Similrly, for Chin we hve Nt b n b Nt. b Ech country hs its own currency, dollrs nd renminbi. The supply of dollrs follows the following eqution: Mt z Mt. And the supply of renminbi follows: Mt b z b Mt. b As usul, we re interested in sttionry monetry equilibri. Do the following: 7
8 () Suppose ech country implements foreign currency controls. Derive the rel rte of return on dollrs nd on renminbi. (b) Explin why the exchnge rte e t hs to be equl to the rtio of vt nd vt b. Suppose tht popultion in Chin grows fster thn in US nd tht both countries expnd their money supply t the sme rte. Will the US dollr pprecite or deprecite over time ginst the renminbi? Explin. (c) Suppose tht ll foreign currency controls were lifted. Demonstrte tht we will not be ble to determine the exchnge rte e t ny more. Discuss. For the rest of the problem, we ssume tht n n b nd tht N N b 00. We will lso ssume tht M M b 600 nd tht the initil money stock is divided eqully mong the initil old genertions in ech country (so z z b ). In ddition, we ssume tht every young person wnts to hold rel money blnces tht re worth 8 units of consumption (so y c y b c b 8). Finlly, the exchnge rte is fixed t ē 2, so dollr cn be exchnged for 2 renminbis. There re no foreign currency controls. (d) Find the vlue (mesured in goods) of dollr nd of renminbi. Wht is the consumption of n old person? (Hint: use the globl money mrket clering condition from prt (c)). (e) Suppose tht every initil old person in US nd in Chin decides to reduce his holdings of Chin s money by 2 renminbi. Every person turns 2 renminbi to the Chinese monetry uthorities wishing to exchnge it for dollr. Assume tht the US monetry uthorities re cooperting with their Chinese counterprts nd re willing to print s mny new dollrs s necessry. Wht will hppen to the totl money stock of dollrs nd renminbi? How will the vlues of ech currency chnge? (f) Now suppose tht the sme sitution hppens s in prt (e), but the US monetry uthorities refuse to cooperte. Insted, the Chinese government decides to honor its pledge for the fixed exchnge rte by txing every old Chinese citizen eqully. Wht will be the vlues of ech currency? How mny goods must ech old Chinese person be txed? How much does n old US citizen consume, nd how much does the old Chinese citizen consume? Who benefits from this policy? Answer () (5 points) With foreign currency controls, ech country hs its own money mrket clering condition. In US we get: which implies: v t+ v t vt N t (y c ) Mt, N t+ (y c ) M t+ N t (y c ) M t N t+ Mt Mt+ Nt n z. 8
9 Similr steps cn be pplied to Chin, nd we will obtin: v b t+ v b t nb z b. (b) (5 points) We must hve e t v t vt b, (28) for people to be willing to hold both currencies in equilibrium. Otherwise, if e t v b t > v t, everyone prefers to hold renminbi insted of dollrs, nd if e t v b t < v t, everyone prefers to hold dollrs insted of renminbi. The evolution of exchnge rte is given by: e t+ e t v t+ vt+ b vt vt b v t+ vt b vt+ b vt n z z b n b, nd if z z b nd n b > n, we hve e t+ e t n n b <, so US dollrs deprecite over time ginst renminbis. Becuse of currency controls, demnd for renminbi grows fster thn demnd for dollrs, nd so every dollr bill becomes reltively less vluble over time. (c) (5 points) Now there will be single worldwide money mrket clering condition: v t M t + v b t M b t N t (y c ) + N b t ( y b c b ), nd given eqution (28), we cn replce v t with e t v b t : e t vt b Mt + vt b Mt b Nt (y c ) + Nt b ( y b c b ) (29) v b t [ et Mt + Mt b ] N t (y c ) + Nt b ( y b c b ), nd we re left with single eqution with two unknowns, v b t nd e t, so there re infinitely mny solutions. The intuition is strightforwrd: in world like this dollrs nd renminbis re perfect substitutes, just like dollr bills printed in Cliforni nd dollr bills printed in Minnesot. So we cnnot determine both the vlues of individul currencies nd their exchnge rtes simultneously. (d) (5 points) Given ll the numbers, nd using eqution (29), we cn obtin: N t (y c ) + N b t ( y b c b ) 00 (8) + 00 (8) 3600, nd since M t M b t 600, we hve v t v b t nd since v t e t v b t, we hve 2v b t + v b t 6 v b t 2 9
10 nd then v t 2 (2) 4. Every old person consumes c 2 v t m t + v b t m b t 4 (3) + 2 (3) 8 units. (We know tht m t m b t 3 since the money supply is divided mong old people eqully.) (e) (5 points) This illustrtes coopertive stbiliztion of exchnge rtes. Since every old person surrenders 2 renminbi, nd there re 200 old people in totl, there re 400 renminbis surrendered. The exchnge rte is fixed t e 2, so people will demnd 200 dollrs in exchnge. Thus the new money supplies re: M t M b t , nd we cn use the sme world money mrket clering condition (29) to determine v t nd v b t : 2v b t (800) + v b t (200) 3600 v b t 2 nd hence v t 4 nd nothing chnges. So coopertive stbiliztion works s complete insurnce for people ginst fluctutions in currency mrkets. Every old person now consumes units, so their welfre is not ffected. c 2 v t m t + v b t m b t 4 (4) + 2 () 8 (f) (0 points) Updte: This prt contins n error. Do not grde it, insted, give everyone 0 points for it. Now we re in world where Chin unilterlly defends the fixed exchnge rte. As before, every old person surrenders 2 renminbi, nd there re 200 old people in totl, so there re 400 renminbis surrendered. The exchnge rte is fixed t e 2, so people will demnd 200 dollrs in exchnge. The Chinese government hs to rise v t (200) of txes in order to honor its pledge. Nturlly, Chin cn only tx its own citizens, so every old Chinese person hs to py tx of τ 200v t 00 2v t. The totl stock of renminbi flls to 200, s in the previous prt. The stock of dollrs is now unffected, however. So the sme condition (29) now implies nd since the exchnge rte is fixed t e t 2, we hve 2v b t (600) + v b t (200) 3600 v b t , v t 2v b t
11 Now the old in US nd Chin consume different mounts. US citizens py no tx, so their consumption is c 2 v t m t + v b t m b t 36 7 (4) () nd they re clerly better-off reltive to the sitution in which both countries cooperted. The vlue of their dollr holdings go up nd they py no tx for this to hppen. Chinese consumers, however, re worse off c b 2 v t m t + v b t m b t τ 36 7 (4) () 36 7 (2) nd you my notice tht consumption of Chinese old went down by precisely the sme mount s the increse of US old consumption.
Math 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
Basic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers
Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from
Vectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
MATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
Small Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
Operations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
Experiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
Small Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
How To Network A Smll Business
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
All pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
Small Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
Reasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
Small Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
Graphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
Week 7 - Perfect Competition and Monopoly
Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
Algebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
Health insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
Integration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
Small Businesses Decisions to Offer Health Insurance to Employees
Smll Businesses Decisions to Offer Helth Insurnce to Employees Ctherine McLughlin nd Adm Swinurn, June 2014 Employer-sponsored helth insurnce (ESI) is the dominnt source of coverge for nonelderly dults
Factoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
Integration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
and thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
Health insurance marketplace What to expect in 2014
Helth insurnce mrketplce Wht to expect in 2014 33096VAEENBVA 06/13 The bsics of the mrketplce As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum
I calculate the unemployment rate as (In Labor Force Employed)/In Labor Force
Introduction to the Prctice of Sttistics Fifth Edition Moore, McCbe Section 4.5 Homework Answers to 98, 99, 100,102, 103,105, 107, 109,110, 111, 112, 113 Working. In the lnguge of government sttistics,
CHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
Warm-up for Differential Calculus
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
How To Set Up A Network For Your Business
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10-438 UNVERSTY OF NOTTNGHAM Discussion
Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
EQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
SPECIAL PRODUCTS AND FACTORIZATION
MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
Firm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
Binary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
Labor Productivity and Comparative Advantage: The Ricardian Model of International Trade
Lbor Productivity nd omrtive Advntge: The Ricrdin Model of Interntionl Trde Model of trde with simle (unrelistic) ssumtions. Among them: erfect cometition; one reresenttive consumer; no trnsction costs,
9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
Lecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
Distributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
Estimating Exchange Rate Exposures:
Estimting Exchnge Rte Exposures: Issues in Model Structure * Gordon M. Bodnr ** Pul H. Nitze School of Advnced Interntionl Studies, The Johns Hopkins University 1740 Msschusetts Avenue NW Wshington, DC
4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
Exponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
Health insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 11/12 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
This paper considers two independent firms that invest in resources such as capacity or inventory based on
MANAGEMENT SCIENCE Vol. 5, No., December 006, pp. 93 99 issn 005-909 eissn 56-550 06 5 93 informs doi 0.87/mnsc.060.0574 006 INFORMS Strtegic Investments, Trding, nd Pricing Under Forecst Updting Jiri
Version 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
Physics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
Section 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
Lecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
Homework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
JaERM Software-as-a-Solution Package
JERM Softwre-s--Solution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock
Optimal Redistributive Taxation with both Labor Supply and Labor Demand Responses
Optiml Redistributive Txtion with both Lbor Supply nd Lbor Demnd Responses Lurence JACQUET NHH Preliminry version Etienne LEHMANN y CREST Bruno VAN DER LINDEN z IRES - UCLouvin nd FNRS Mrch 28, 2011 Abstrct
Or more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
Optiml Control of Seril, Multi-Echelon Inventory (E&I) & Mixed Erlng demnds
Optiml Control of Seril, Multi-Echelon Inventory/Production Systems with Periodic Btching Geert-Jn vn Houtum Deprtment of Technology Mngement, Technische Universiteit Eindhoven, P.O. Box 513, 56 MB, Eindhoven,
www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
Project Recovery. . It Can Be Done
Project Recovery. It Cn Be Done IPM Conference Wshington, D.C. Nov 4-7, 200 Wlt Lipke Oklhom City Air Logistics Center Tinker AFB, OK Overview Mngement Reserve Project Sttus Indictors Performnce Correction
Rotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
Network Configuration Independence Mechanism
3GPP TSG SA WG3 Security S3#19 S3-010323 3-6 July, 2001 Newbury, UK Source: Title: Document for: AT&T Wireless Network Configurtion Independence Mechnism Approvl 1 Introduction During the lst S3 meeting
The Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
Pre-Approval Application
Pre-Approvl Appliction In tody s rel estte mrket, Pre-Approved mortgge provides you the buyer with powerful tool in the home purchse process! Once you hve received your Pre-Approvl, you cn shop for your
FUNDING OF GROUP LIFE INSURANCE
TRANSACTIONS OF SOCIETY OF ACTUARIES 1955 VOL. 7 NO. 18 FUNDING OF GROUP LIFE INSURANCE CHARLES L. TROWBRIDGE INTRODUCTION T RADITIONALLY the group life insurnce benefit hs been finnced on yerly renewble
Lectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
Applications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
AREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
2001 Attachment Sequence No. 118
Form Deprtment of the Tresury Internl Revenue Service Importnt: Return of U.S. Persons With Respect to Certin Foreign Prtnerships Attch to your tx return. See seprte instructions. Informtion furnished
DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
Industry and Country Effects in International Stock Returns
Industry nd Country ffects in Interntionl Stock Returns Implictions for sset lloction. Steven L. Heston nd K. Geert Rouwenhorst STVN L. HSTON is ssistnt professor of finnce t the John M. Olin School of
Pay over time with low monthly payments. Types of Promotional Options that may be available: *, ** See Page 10 for details
With CreCredit... Strt cre immeditely Py over time with low monthly pyments For yourself nd your fmily Types of Promotionl Options tht my be vilble: Not ll enrolled helthcre prctices offer ll specil finncing
Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
Unit 29: Inference for Two-Way Tables
Unit 29: Inference for Two-Wy Tbles Prerequisites Unit 13, Two-Wy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl
2. Transaction Cost Economics
3 2. Trnsction Cost Economics Trnsctions Trnsctions Cn Cn Be Be Internl Internl or or Externl Externl n n Orgniztion Orgniztion Trnsctions Trnsctions occur occur whenever whenever good good or or service
Section 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
Data replication in mobile computing
Technicl Report, My 2010 Dt repliction in mobile computing Bchelor s Thesis in Electricl Engineering Rodrigo Christovm Pmplon HALMSTAD UNIVERSITY, IDE SCHOOL OF INFORMATION SCIENCE, COMPUTER AND ELECTRICAL
Regular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
MODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
VoIP for the Small Business
Reducing your telecommunictions costs VoIP (Voice over Internet Protocol) offers low cost lterntive to expensive trditionl phone services nd is rpidly becoming the communictions system of choice for smll
19. The Fermat-Euler Prime Number Theorem
19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT
COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified
Enterprise Risk Management Software Buyer s Guide
Enterprise Risk Mngement Softwre Buyer s Guide 1. Wht is Enterprise Risk Mngement? 2. Gols of n ERM Progrm 3. Why Implement ERM 4. Steps to Implementing Successful ERM Progrm 5. Key Performnce Indictors