Start with a general production function showing output (Y) as a function of capital (K) and labor (N):
|
|
- Meredith Brown
- 7 years ago
- Views:
Transcription
1 Homewor 1 Le Cabrera Describe the effects o output per worer () ad capital per worer () b chages i savigs rate (s), labor growth rate (), depreciatio rate (δ), ad techical progress (A). There should be eight multipliers. Determie if each is either positive or egative ad show the results i a diagram. For the first three exogeous variables (s,, δ), use a geeral fuctio f() with costat returs to scale. For techical progress (A), use the Cobb-Douglass productio fuctio = f() = A Start with a geeral productio fuctio showig output (Y) as a fuctio of capital (K) ad labor (N): Y = F( K, N) (1) Note that output per worer () is equal to output divided b the umber of worers: Y F( K, N) = = (2) N N Sice the productio fuctio has costat returs to scale, we ca rewrite this as: F( K, N) K N = = F, N N N (3) Note that the secod argumet to the productio fuctio is ow 1 ad the first argumet is simpl the capital per worer (). Therefore, equatio (3) ca be rewritte as a ew fuctio of ol oe variable: This fuctio is assumed to have dimiishig returs: = f (4) f() Figure 1. Dimiishig Returs 1 of 8
2 Before goig o, realize that there are three other "iputs" to this model, but all are treated as costat (i.e., ot explicitl part of the ew fuctio). These iclude the savigs rate (s), the growth rate of labor (), ad the depreciatio rate (δ). These factor ito the model whe we loo at how K ad N chage over time: dk = I δk (5) dn = N (6) These formulas should be ituitive. Capital (K) icreases b the amout of ew ivestmet (I) ad decreases b the amout of depreciatio, which is simpl the depreciatio rate times the curret amout of capital (i.e., δk). The labor force grows b the labor growth rate times the curret umber of worers (i.e., N). Notice that equatio (5) ca be rewritte usig the savigs rate (s) ad the fact that I = sy: dk = sy δk (7) What we reall wat to loo at, however, is how capital per worer () chages over time. Note that is a fuctio of both K ad N, so we must tae the total derivative (the sum of all the partial derivatives; Simo ad Blume p.311), made slightl more difficult b also havig to appl the chai rule: d This equatio simplifies to: d dk d dn 1 K = = ( sy δ K ) ( N ) (8) 2 dk dn N N d Y K = s ( δ ) (9) N N Which ca further be simplified b usig the output per worer ( = f()) ad capital per worer (): d ( ) ( δ ) = sf (10) Now to determie how savigs, labor growth, ad depreciatio rates (s,, ad δ, respectivel) affect ad, we eed to cosider the stead state sceario. That is, the poit where is ot chagig over time: 2 of 8
3 d ( ) = 0 = sf δ (11) Note that at this poit, the icrease i capital per worer (sf(), also called actual ivestmet) is equal to the decrease i capital per worer (( δ), also called breaeve ivestmet). This decrease is made up of depreciatio ad capital trasferred to ew worers. The poit where the are equal (i.e., actual ivestmet equals brea-eve ivestmet) is deoted b * ad is called the balaced growth path (Romer p.17). Graphicall: * f() ( δ) sf() Figure 2. Brea Eve Ivestmet To loo at the chage i s,, ad δ, we have to tae the total derivative of equatio (11). (Recall this meas we sum all the partial derivatives.) ( d ) ( d ) ( d ) ( d ) s d ( δ ) d f ds d dδ 0 sf ' = (12) Now rearrage terms to fid d. The terms that are multiplied b ds, d, ad dδ are the multipliers (i.e., the show how chages whe s,, ad δ chage, respectivel). [( δ ) sf ] d = f ds d dδ * ' (13) δ d f = δ sf ' ds d δ sf '( ) δ sf ' dδ (14) From here we ca fid how these iputs effect output per worer () b taig the derivative of equatio (4): Pluggig equatio (14) ito equatio (15) results i: d = f 'd (15) 3 of 8
4 f d = f ' δ sf ' f ' ds δ sf ' f ' d δ sf ' dδ (16) Let's loo at how each factor affected ad : Savigs Rate (s) Multiplier: ( ) sf '( ) f δ > 0 Multiplier: ( ) f ' δ sf '( ) f > 0 I order to determie the sigs of these multipliers mathematicall, we eed to determie whether the umerators ad deomiators are positive or egative. Sice both share the same deomiator, we should start there. Note that the first part ( δ) is simpl the slope of the brea-eve ivestmet curve (i.e., the straight lie i Figure 2). The secod part (sf '()) is the slope of the actual ivestmet curve (i.e., the shallower curve i Figure 2). I the area aroud the balaced growth path (i.e., *), this differece is positive because the straight lie is icreasig faster tha the curved oe. Therefore, both multipliers have positive deomiators. It should be fairl obvious that both multipliers also have positive deomiators sice the productio fuctio, f(x), is ever egative ad the slope, although dimiishig, is alwas positive. With positive umerators ad positive deomiators, both multipliers are positive. This maes sese, because icreased savigs rate should result i more capital per worer, hece more output per worer. The effects o ad ca be show graphicall as well: 2 1 f() ( δ) s 2 f() s 1 f() 1 2 Figure 3. Icreased Savigs Rate Labor Growth Rate () Multiplier: δ sf ' < 0 Multiplier: f ' δ sf ' < 0 Here agai we wat to loo at the umerators ad deomiators separatel first. Both multipliers share the same deomiator which is positive as discussed i the previous 4 of 8
5 sectio. Both umerators are egative. This is clear for the Multiplier sice is positive ad it has a egative sig i the multiplier. For the Multiplier, - is multiplied b f '() which is positive ad the resultig product is egative. Ituitivel, this result maes sese because if the labor growth rate icreases (all else beig equal), there will be less capital per worer (i.e., ) ad less output per worer (i.e., ). The effects o ad ca be show graphicall as well: 1 2 f() ( 2 δ) ( 1 δ) s 1 f() 2 1 Figure 4. Icreased Labor Growth Rate Depreciatio Rate (δ) Multiplier: δ sf ' < 0 Multiplier: f ' δ sf ' < 0 A icrease i the depreciatio rate has the exact same effect as a icrease i the labor growth rate. Eve the graphs loo the same, except for the fact that δ is chagig istead of : 1 2 f() ( δ 2 ) ( δ 1 ) s 1 f() 2 1 Figure 5. Icreased Depreciatio Rate Techical Progress (A) We ow tur our attetio to the impact of techical progress (A), b looig at a Cobb- Douglass productio fuctio = f() = A. This fuctio is substituted ito equatio (11) ad we follow alog similar reasoig we used before to fid d i terms of da: 5 of 8
6 d ( δ ) = 0 = sa (17) Although it is ot ecessar at this poit to also loo at the chage i s,, ad δ, we might as well i order to double chec the geeric multipliers geerated earlier. Taig the total derivative of equatio (17) ields: sa ( d ) ( d ) ( d ) ( d ) ( d ) s δ A ( δ ) d A ds d dδ s da = 0 d (18) Now rearrage terms to fid d. The terms that are multiplied b ds, d, dδ, ad da are the multipliers. [( δ ) sa ] d = A ds d dδ s da d = A ds d dδ δ sa δ sa δ sa s δ sa da (19) (20) Note that the multipliers for s,, ad δ are the same as the oes derived earlier whe ou substitute f() = A ad f '() = A -1. There is ow a additioal cosideratio before movig o to fid d. We have to treat as a fuctio of both A ad so we must oce agai tae a total derivative: d = f ' ' d f da = A d da A (21) Pluggig equatio (20) ito equatio (21) results i: 2 2 d = A A A ds d dδ δ sa δ sa δ sa 2 sa da δ sa (22) 6 of 8
7 Summarizig the impact of A: s Multiplier: > 0 δ sa 2 sa Multiplier: > 0 δ sa ( δ ) = δ sa Followig the same logic used whe looig at the multipliers for s, the deomiators of both multipliers here are positive. The umerators are also positive sice each idividual term is positive. The multiplier for adds the term which is also positive. Ituitivel, this result maes sese because if there is techological progress, we would expect the amout of capital per worer ad the output per worer to icrease. Graphicall this result is slightl differet tha the others because the chage i A directl effects f() so there are two shifts: 2 1 f 2 () f 1 () ( δ) sf 2 () sf 1 () 1 2 Figure 6. Icreased Techological Progress Documetatio This was compiled predomiatl from class otes. Where oted the course text (Advaced Macroecoomics b Romer) ad Mathematics for Ecoomists (Carl P. Simo ad Lawrece Blume) were used. Specific help was received from Professor Bomberger i class. He cleared up where we were startig from to get d/ = 0. He also clarified some of the poits o techological progress. Guille Sabbioi ad I discussed figurig out wh the deomiators were positive. I ew from class that the were, but did't recall wh. Guille also caught m error i ot taig the derivative of with respect to both ad A i the last sectio. I double checed that with Professor Bomberger. 7 of 8
8 Coclusios from Solow Model Exogeous - "outside the model"; iputs to the model; values tae as give ad ot determied b the model; for Solow: ds & d Edogeous - "iside the model"; outputs of the model; values determied b the model; for Solow: d & d Stead State - evetuall growth i output, growth i capital, ad growth i labor are the same ( Y/Y = K/K = N/N = ) Icrease i savigs rate (s ) - growth of output ad capital icreases for a while, but evetuall levels off at origial rate (); but ow have a higher ad tha would've had before (oe time icrease i stadard of livig) NOTE: icrease i is modest (e.g., 50% icrease i s ields 14% icrease i ) ad is slow to materialize Icrease i labor growth rate ( ) - results i faster growth of output ad capital (equal to the ew ), but lower output per worer (Y/N ) ad lower stadard of livig Techological Progress (A) - A is idex of techological progress: A T = ( 1 g) AT Several was to icorporate it: Y = F(K,N) - example: AK N 1- Y = F(K,AN) - labor-augmetig techical chage; preferred for coveiece although mathematicall, all are equivalet for Cobb-Douglas fuctio Y = F(AK,N) - capital-augmetig techical chage Y K Now = ad =, where AN = effective labor AN AN d = sf ( δ g) Y Differece From No A i Model - = g (used to equal ) Y 8 of 8
Soving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationTHE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationMath 113 HW #11 Solutions
Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationCHAPTER 11 Financial mathematics
CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula
More informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationhp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationA Recursive Formula for Moments of a Binomial Distribution
A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,
More information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationQuestion 2: How is a loan amortized?
Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationMann-Whitney U 2 Sample Test (a.k.a. Wilcoxon Rank Sum Test)
No-Parametric ivariate Statistics: Wilcoxo-Ma-Whitey 2 Sample Test 1 Ma-Whitey 2 Sample Test (a.k.a. Wilcoxo Rak Sum Test) The (Wilcoxo-) Ma-Whitey (WMW) test is the o-parametric equivalet of a pooled
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationTHE TIME VALUE OF MONEY
QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationZ-TEST / Z-STATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
Z-TEST / Z-STATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large T-TEST / T-STATISTIC: used to test hypotheses about
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationLearning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More informationPage 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville
Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What
More informationThis document contains a collection of formulas and constants useful for SPC chart construction. It assumes you are already familiar with SPC.
SPC Formulas ad Tables 1 This documet cotais a collectio of formulas ad costats useful for SPC chart costructio. It assumes you are already familiar with SPC. Termiology Geerally, a bar draw over a symbol
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationLecture 2: Karger s Min Cut Algorithm
priceto uiv. F 3 cos 5: Advaced Algorithm Desig Lecture : Karger s Mi Cut Algorithm Lecturer: Sajeev Arora Scribe:Sajeev Today s topic is simple but gorgeous: Karger s mi cut algorithm ad its extesio.
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationFM4 CREDIT AND BORROWING
FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer
More informationHow to use what you OWN to reduce what you OWE
How to use what you OWN to reduce what you OWE Maulife Oe A Overview Most Caadias maage their fiaces by doig two thigs: 1. Depositig their icome ad other short-term assets ito chequig ad savigs accouts.
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationFrance caters to innovative companies and offers the best research tax credit in Europe
1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public
More informationForecasting techniques
2 Forecastig techiques this chapter covers... I this chapter we will examie some useful forecastig techiques that ca be applied whe budgetig. We start by lookig at the way that samplig ca be used to collect
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationTradigms of Astundithi and Toyota
Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationComplex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have
Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More information