ENGS 41. SUSTAINABILITY and NATURAL RESOURCE MANAGEMENT. The Cohort. (Lynch book, Chapter 4) Benoit Cushman-Roisin January 2018
|
|
- Helen Barker
- 5 years ago
- Views:
Transcription
1 ENGS 41 SUSTAINABILITY and NATURAL RESOURCE MANAGEMENT The Cohor (Lynch book, Chaper 4 Benoi Cushman-Roisin January 2018 Wha is a cohor? A cohor consiss o a group o individuals who ogeher progress hrough sages. One cohor goes hrough muliple sages. You, he sudens, make cohors: Class o 2021, now 1 s -year, will be sophomores, ec. Class o 2020, now sophomores, will be juniors, hen seniors Class o 2019, were sophomores, now juniors, will be seniors Class o 2018, were sophomores, hen juniors, now seniors 1
2 Fish cohor: Fish sar as eggs and hen evolve as larvae, ry, juveniles o evenually become adul ish. One can urher disinguish muliple sages in each o hese sages, like juvenile-1, juvenile-2, adul-1, adul-2, adul-3 o he degree o desired biological precision. Each year (or season, a new cohor is generaed, which co-lives wih he previous cohors. Time is he variable ha makes he individuals progress inside heir respecive cohor. There are oher biological sysems besides isheries, like crops and oress. Depending on he biological sysem and is managemen, he recruimen in each cohor can be eiher endogenous or exogenous. Endogenous recruimen (in he wild, unconrolled Example: Number o ish larvae proporional o adul populaion. Exogenous recruimen (humanly conrolled Examples: Nursery plans are uprooed and replaned somewhere else. In aquaculure, juveniles are reained in numbers needed. The ime o recruimen is sill se by he biology, bu he amoun o recruimen ino he cohor is decided exernally wihou relaion o he number o individuals a a laer sage. 2
3 Harvesing versus cach: During a ishing expediion, one akes rom various cohors a he same ime. oo young maure enough cach Basic decisions in harvesing are: 1. A which sage o begin? 2. How hard o work a i? Beveron-Hol cohor model: (Lynch book, Secion We deine he ollowing variables or a cohor: = age since recruimen ino he cohor W( = weigh o an individual a age N( = number o individuals a age B( = W( N( = biomass a age R = recruimen ino he cohor N(=0 = R moraliy (age independen = harvesing moraliy = age a which harvesing begins H( = cumulaive harves, rom up o age H oal = H( h = harves per recrui = H oal /R. Survival & Harves dn dn N ( N dh dh 0 B WN 0 3
4 Prior o harvesing: 0 < < Number o individuals dn N N(0 R N Re Corresponding biomass B ( WN ( ( RW ( e Biomass increases, reaches a maximum and hen decreases db dw dw R e RW( e R W e db dw 0 or W Deine p as he age a which biomass is greaes. Then, in he presence o harvesing: < Number o individuals dn ( N N Re e N ( Re Corresponding biomass Harves Toal harves ( ( B ( WN ( ( RWe ( e ( ( ( ( H ( B ( Re We ( ( ( H Re W( e oal Hoal ( ( h e W( e h(, R 4
5 Variaion in sar ime Variaion in Thus, he harves per recrui (harves inensiy is: Hoal ( ( h (, e W( e R Basic decisions in harvesing are: 1. A which sage o begin?? 2. How hard o work a i?? Mahemaically, he maximum is obained or = p and =. I is: Bp hideal h( p, R Cu all hese rees as soon as hey ge big and beore hey die! 5
6 A oal cach a he peak o biomass is usually no possible ( <. To maximize he harves wih inie, one needs o sar beore peak: p h (, ( ( 0 or h (, e W( e This is called he Eumeric Harves. We denoe i h max (. Foreser s Crierion: (Lynch, Secion You are a oreser and you manage several parcels o ores. When a parcel reaches a cerain age, say, you cu all is rees in one operaion and hen replan wih seedlings. Aer ime, he seedlings have grown ino bigger rees, and you cu again. You repea he procedure on his parcel over and over. (Over he years ha his paricular parcel akes o regrow, you deal wih he oher parcels in order o spread your income over ime. This is why you parcel ou he ores. A each cu, since you ake all he wood a he ime in one scoop, he amoun you ge is B( Since you repea his every ime inerval, your harves averaged over ime or his parcel is: B( H 6
7 Naurally, you seek o maximize your harvesing o maximize your revenue. B( Thus, you seek he maximum o H Since he age a which you cu is or you o choose, i is he variable wih respec o which you seek he maximum: dh 1 db ( B ( db ( B ( p Noe ha his sraegy makes you harves beore peak biomass. Yes, you harves early bu you harves more requenly! Wai a minue, hough! This was in order o maximize he biomass harvesed. Wha abou he revenue? Sequenial cohor The Fausman Roaion: (Lynch, Secion Neglecing he cos o harvesing or he momen (or simply subracing a ixed racion o he selling price, he income is: Need o wai ime beore he irs round, beore second round, ec. r 2r 3r pb( e pb( e pb( e 1 pb( r e 1 Maximum income or such ha This is he Fausman Crierion. db( 1 rb( rb( r 1 e 7
8 I reseeding comes a a ixed cos, hen he cos o he successive seedings is: C C C C C r r 2r 3r cce ce ce c e c 1 The proi is: Cos o iniial seeding; all oher reseedings immediaely ollow he sequenial harvess. C ( C ( C ( C pb( c c r e 1 Maximizaion occurs or chosen such ha db( c 1 rb( p e r 1 8
9 Example: (Lynch, Secion 4.2 Le s work a case or which we know he growh rae: Wih his rae o growh, he weigh o an individual over ime is: W ( W 1e g max dw g W W max During his ime, he number o individuals decreases gradually due o some moraliy according o: N( Re The oal biomass a age is: g B( W( N( RW 1e e The peak biomass B p occurs a p wih: g g p e Bp RWmax g g g max g Saring a ime wih eor, he harves h per recrui is ound o be (los o algebra here! 1 1 g h (, Wmax e e g A ixed eor, he harves is maximized i i begins a ime given by: (los o algebra here, oo! opimal 1 g p ln g p 9
10 Economic Harvesing: (Lynch, Secion 4.3 So ar, excep once, we only considered biological aspecs and ried o maximize he amoun harvesed. There were no consideraions o coss and revenues, and o value discouning over ime. Cos o eor: Cos c Revenue rom harves: dh Revenue p p B [Recall: In his chaper, harves H is cumulaive.] Ne: Proi = Revenue Cos p Bc c p B p 10
11 The proi c p B p is posiive only or B c B p 0 p BB 0 Growh prior o harves Decrease during harvesing Harves is inerruped when biomass drops below proiable level. Pos-harves moraliy T Harves sops once B drops o B 0 Now considering ha harvesing akes ime and ha uure income needs o be discouned. Firs, i eor is ininie (hanks o magical echnology!, harves is insananeous bu a ime in he uure; p B( B e 0 r This is maximized when is chosen such ha: d db db 0 r B B 0 r B( B 0 0 The economic harveser does no wai or B o peak. As soon as biological growh is less han inancial growh, B is harvesed (in an insan and sold. The money is pu in he bank and grows here. Increasing ineres rae r hasens he harvesing ime. 11
12 The siuaion is obviously more complicaed when he harves is spread over ime because echnology (, assumed consan allows only a cerain pace. The proi is now calculaed as an inegral over ime: ( r T 0 ( 0 r p B B e p B B e insananeous harves exended harves B0 c p The maximum o occurs when he ollowing relaion holds: T r B( e B( B e 0 r Mixed cohors: (Lynch, Secion During a ishing expediion, one akes rom various cohors a he same ime. oo young maure enough cach We assume: 1. A new cohor sars every year. 2. Each cohor lives muliple years, hus overlapping one anoher. 3. The various cohors live peaceully among one anoher. 4. There is no overcrowding. We ask he same quesions as or a single cohor: 1. A which age do we sar harvesing?? 2. How inensely do we harves?? 12
13 oo young maure enough cach Harvesing across cohors in a single year is equivalen o whole-lie harvesing o a single cohor. Accumulaing across cohors is he same as accumulaing across ime or a single cohor. We hus have a sysem ha will reurn, in a single year, he whole-lie harves o one cohor. This is a biological annuiy wihou urher inancial discouning: p Rh(, c Thus, he proi o one cach is: p Rh(, c I he age o he younges individuals harvesed is chosen o opimize he harves, he harves per recrui h(,m is he eumeric harves, h max (. In his case, p Rh ( max c Eor ha yields he maximum proi No need o make more eor; almos everyhing is already being caugh. 13
) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More informationN t = N o e rt. Plot N t+1 vs. N t. Recruits Four Desirable Properties of S-R Relationships
Populaion Models in Fisheries Models for fish populaions are similar as hose used for birds and mammals. Some differen applicaions have been developed specific o fisheries. Sock Recrui Relaionships Influence
More informationN H. be the number of living fish outside area H, and let C be the cumulative catch of fish. The behavior of N H
ALTRNATV MODLS FOR CPU AND ABUNDANC Fishing is funamenally a localize process. Tha is, fishing gear operaing in a paricular geographic area canno cach fish ha are no in ha area. Here we will evelop wo
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationLecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model
Lecure Noes 3: Quaniaive Analysis in DSGE Models: New Keynesian Model Zhiwei Xu, Email: xuzhiwei@sju.edu.cn The moneary policy plays lile role in he basic moneary model wihou price sickiness. We now urn
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationIntermediate Macro In-Class Problems
Inermediae Macro In-Class Problems Exploring Romer Model June 14, 016 Today we will explore he mechanisms of he simply Romer model by exploring how economies described by his model would reac o exogenous
More informationSeminar 4: Hotelling 2
Seminar 4: Hoelling 2 November 3, 211 1 Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a
More informationAge (x) nx lx. Age (x) nx lx dx qx
Life Tables Dynamic (horizonal) cohor= cohor followed hrough ime unil all members have died Saic (verical or curren) = one census period (day, season, ec.); only equivalen o dynamic if populaion does no
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationCooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.
Cooperaive Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS Augus 8, 213 8:45 a.m. o 1: p.m. THERE ARE FIVE QUESTIONS ANSWER ANY FOUR OUT OF FIVE PROBLEMS.
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More information( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor
8. Economic growh 8.. Producion funcion: Capial as producion facor Y = α N Y (, ) = F K N Diminishing marginal produciviy of capial and labor: (, ) F K L F K 2 ( K, L) K 2 (, ) F K L F L 2 ( K, L) L 2
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationWelcome Back to Physics 215!
Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion
More informationUSP. Surplus-Production Models
USP Surplus-Producion Models 2 Overview Purpose of slides: Inroducion o he producion model Overview of differen mehods of fiing Go over some criique of he mehod Source: Haddon 2001, Chaper 10 Hilborn and
More informationCalculus Tricks #1. So if you understand derivatives, you ll understand the course material much better. a few preliminaries exponents
Calculus Tricks # Eric Doviak Calculus is no a pre-requisie or his course. However, he oundaions o economics are based on calculus, so wha we ll be discussing over he course o he semeser is he inuiion
More information3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationThis document was generated at 7:34 PM, 07/27/09 Copyright 2009 Richard T. Woodward
his documen was generaed a 7:34 PM, 07/27/09 Copyrigh 2009 Richard. Woodward 15. Bang-bang and mos rapid approach problems AGEC 637 - Summer 2009 here are some problems for which he opimal pah does no
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationSterilization D Values
Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationADVANCED MATHEMATICS FOR ECONOMICS /2013 Sheet 3: Di erential equations
ADVANCED MATHEMATICS FOR ECONOMICS - /3 Shee 3: Di erenial equaions Check ha x() =± p ln(c( + )), where C is a posiive consan, is soluion of he ODE x () = Solve he following di erenial equaions: (a) x
More informationOn Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationThe average rate of change between two points on a function is d t
SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope
More informationIn Game Theory and Applications, Volume X WITH MIGRATION: RESERVED TERRITORY APPROACH. Vladimir V. Mazalov. Anna N. Rettieva
In Game Theory and Applicaions, Volume X Edied by L.A. Perosjan and V.V. Mazalov, pp.97{10 ISBN 1-59033-34-X c 004 Nova Science Publishers, Inc. A FISHERY GAME MODEL WITH MIGRATION: RESERVED TERRITORY
More informationKalman filtering for maximum likelihood estimation given corrupted observations.
alman filering maimum likelihood esimaion given corruped observaions... Holmes Naional Marine isheries Service Inroducion he alman filer is used o eend likelihood esimaion o cases wih hidden saes such
More informationLecture #6: Continuous-Time Signals
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions
More information72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s
More informationSuggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class
EC 450 Advanced Macroeconomics Insrucor: Sharif F Khan Deparmen of Economics Wilfrid Laurier Universiy Winer 2008 Suggesed Soluions o Assignmen 4 (REQUIRED) Submisson Deadline and Locaion: March 27 in
More informationnot to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling?
256 MATHEMATICS A.2.1 Inroducion In class XI, we have learn abou mahemaical modelling as an aemp o sudy some par (or form) of some real-life problems in mahemaical erms, i.e., he conversion of a physical
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationAdvanced Organic Chemistry
Lalic, G. Chem 53A Chemisry 53A Advanced Organic Chemisry Lecure noes 1 Kineics: A racical Approach Simple Kineics Scenarios Fiing Experimenal Daa Using Kineics o Deermine he Mechanism Doughery, D. A.,
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationA Dynamic Model of Economic Fluctuations
CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model
More informationMath 115 Final Exam December 14, 2017
On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):
More informationF This leads to an unstable mode which is not observable at the output thus cannot be controlled by feeding back.
Lecure 8 Las ime: Semi-free configuraion design This is equivalen o: Noe ns, ener he sysem a he same place. is fixed. We design C (and perhaps B. We mus sabilize if i is given as unsable. Cs ( H( s = +
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More informationLecture 3: Solow Model II Handout
Economics 202a, Fall 1998 Lecure 3: Solow Model II Handou Basics: Y = F(K,A ) da d d d dk d = ga = n = sy K The model soluion, for he general producion funcion y =ƒ(k ): dk d = sƒ(k ) (n + g + )k y* =
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationExamples of Dynamic Programming Problems
M.I.T. 5.450-Fall 00 Sloan School of Managemen Professor Leonid Kogan Examples of Dynamic Programming Problems Problem A given quaniy X of a single resource is o be allocaed opimally among N producion
More informationSection 7.4 Modeling Changing Amplitude and Midline
488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves
More informationUCLA: Math 3B Problem set 3 (solutions) Fall, 2018
UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 This problem se concenraes on pracice wih aniderivaives. You will ge los of pracice finding simple aniderivaives as well as finding aniderivaives graphically
More informationaa aa aa 1 aa, ab aa 1 aa,
Tex S: odel equaions or edusa sochasic discree-ime ramework: In he main ex we described a sochasic ramework or modeling he spread o edusa hrough a randomly maing populaion; however we le ou several equaions
More informationExplaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
More informationProblem Set #3: AK models
Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal
More informationDisplacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
More informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
More informationpe pt dt = e pt Probabilty of death given survival till t : pe pt = p Expected life at t : pe(s t)p ds = e (s t)p t =
BLANCHARD Probabiliy of Deah: π () = pe p ; Probabily of living ill : Ω () = pe p d = e p Probabily of deah given survival ill : pe p = p e p Expeced life a : (s ) pe (s )p ds = p 1 Populaion normalized
More informationReserves measures have an economic component eg. what could be extracted at current prices?
3.2 Non-renewable esources A. Are socks of non-renewable resources fixed? eserves measures have an economic componen eg. wha could be exraced a curren prices? - Locaion and quaniies of reserves of resources
More informationSolutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationT L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB
Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal
More informationEconomics 8105 Macroeconomic Theory Recitation 6
Economics 8105 Macroeconomic Theory Reciaion 6 Conor Ryan Ocober 11h, 2016 Ouline: Opimal Taxaion wih Governmen Invesmen 1 Governmen Expendiure in Producion In hese noes we will examine a model in which
More informationFinal Exam. Tuesday, December hours
San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all
More informationTime series Decomposition method
Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,
More informationOsipenko Denis, Retail Risk Management, Raiffeisen Bank Aval JSC, Kiev, Ukraine. Credit Scoring and Credit Control XII conference August 24-26, 2011
Osipenko enis Reail Risk Managemen Raiffeisen Bank Aval JSC Kiev Ukraine Credi Scoring and Credi Conrol XII conference Augus - By he reason of risks inerpeneraion: Credi Risk => osses => Balance iquidiy
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationOnline Appendix for "Customer Recognition in. Experience versus Inspection Good Markets"
Online Appendix for "Cusomer Recogniion in Experience versus Inspecion Good Markes" Bing Jing Cheong Kong Graduae School of Business Beijing, 0078, People s Republic of China, bjing@ckgsbeducn November
More informationA First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationChapter 2. Models, Censoring, and Likelihood for Failure-Time Data
Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based
More informationMath 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007
NAME: Suden ID #: QUIZ SECTION: Mah 111 Miderm I, Lecure A, version 1 -- Soluions January 30 h, 2007 Problem 1 4 Problem 2 6 Problem 3 20 Problem 4 20 Toal: 50 You are allowed o use a calculaor, a ruler,
More informationChapter 3 Common Families of Distributions
Chaer 3 Common Families of Disribuions Secion 31 - Inroducion Purose of his Chaer: Caalog many of common saisical disribuions (families of disribuions ha are indeed by one or more arameers) Wha we should
More informationAPPM 2360 Homework Solutions, Due June 10
2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More information2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance
Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion
More information1 Review of Zero-Sum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of Zero-Sum Games Las ime we inroduced a mahemaical model for wo player zero-sum games. Any
More informationSection 4.4 Logarithmic Properties
Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
More informationFishing limits and the Logistic Equation. 1
Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More information(MS, ) Problem 1
MS, 7.6.4) AKTUAREKSAMEN KONTROL I FINANSIERING OG LIVSFORSIKRING ved Københavns Universie Sommer 24 Skriflig prøve den 4. juni 24 kl..-4.. All wrien aids are allowed. The wo problems of oally 3 quesions
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationNEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
More informationEE100 Lab 3 Experiment Guide: RC Circuits
I. Inroducion EE100 Lab 3 Experimen Guide: A. apaciors A capacior is a passive elecronic componen ha sores energy in he form of an elecrosaic field. The uni of capaciance is he farad (coulomb/vol). Pracical
More informationin Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology
Risk and Saey in Engineering Pro. Dr. Michael Havbro Faber ETH Zurich, Swizerland Conens o Today's Lecure Inroducion o ime varian reliabiliy analysis The Poisson process The ormal process Assessmen o he
More informationNon-uniform circular motion *
OpenSax-CNX module: m14020 1 Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform
More informationIntroduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
More informationF.LE.A.4: Exponential Growth
Regens Exam Quesions F.LE.A.4: Exponenial Growh www.jmap.org Name: F.LE.A.4: Exponenial Growh 1 A populaion of rabbis doubles every days according o he formula P = 10(2), where P is he populaion of rabbis
More informationOnline Convex Optimization Example And Follow-The-Leader
CSE599s, Spring 2014, Online Learning Lecure 2-04/03/2014 Online Convex Opimizaion Example And Follow-The-Leader Lecurer: Brendan McMahan Scribe: Sephen Joe Jonany 1 Review of Online Convex Opimizaion
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationEE650R: Reliability Physics of Nanoelectronic Devices Lecture 9:
EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he
More informationT. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 2011 EXAMINATION
ECON 841 T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 211 EXAMINATION This exam has wo pars. Each par has wo quesions. Please answer one of he wo quesions in each par for a
More information