15.023J / J / ESD.128J Global Climate Change: Economics, Science, and Policy Spring 2008

Size: px
Start display at page:

Download "15.023J / J / ESD.128J Global Climate Change: Economics, Science, and Policy Spring 2008"

Transcription

1 MIT OpenCourseWare hp://ocw.mi.edu J / J / ESD.128J Global Climae Change: Economics, Science, and Policy Spring 2008 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.

2 Massachuses Insiue of Technology ESD.128 Climae Change: Economics, Science and Policy Problem Se #2 MODELING ECONOMIC GROWTH AND EMISSIONS Due Monday, 17 March THE ASSIGNMENT In his assignmen you will apply a spreadshee-based model of global CO 2 emissions o prepare 100-year projecions of economic growh and carbon emissions, and o carry ou sudies of emissions resricion. The purpose of he exercise is o gain insigh ino he ways such models are consruced and applied. The model used here is simplified by ignoring he influence of governmen axaion and expendiure, and inernaional rade, and by implemening a simple srucure of producion and inpu pricing. The model is laid ou in Secion 2 along wih parameer values. A emplae wih mos of he equaions already wrien in is provided for download on he course websie. Columns A hrough Q are used in his homework se. You will work wih boh recursive-dynamic (myopic) and forward looking versions. Analyses o be conduced are described in Secion 3. Secion 4 describes how he emplae workshee was se up, and Secion 5 provides some workshee hins. There are nine pars: A1, A2, B1, B2, B3, C1, C2, C3 and D. You may find i convenien o include prin-ous of he solved workshees in your answers. 2. THE EMISSIONS PREDICTION MODEL 1. The Model of Economic Growh and Emissions Naional oupu is modeled as a funcion of hree facors: labor, capial and energy. The equaions for labor and capial are similar o hose discussed in class on 3 March. Carbon dioxide emissions are modeled as proporional o energy consumpion, and he emissionsmiigaion policy insrumen is direc conrol of he level of energy use. Analysis of oher measures ha migh be used in emissions conrol, such as axes and subsidies, is no possible wih his formulaion. Economic variables are saed on an annual basis ($10 12 /yr). The model is solved on a 10-year ime sep.

3 Homework #2 p GDP Growh World GDP is represened by a single good, y, which is modeled as a funcion of inpus of capial, k, labor, l, and energy, e. The unis of y are rillion (10 12 ) $US. The oupu of he economy is represened by a consan-elasiciy-of-subsiuion producion funcion as follows, ρ y = a α k KLE ρ + β l KLE ρ [ + (1 α β) e KLE where he elasiciy of subsiuion, σ KLE, is 1 σ KLE =. 1 ρ KLE The labor force grows a an annual rae λ, 0 (1 + λ 1/ρ KLE (1) l = l ). (2) The capial sock changes over ime as a resul of depreciaion and new invesmen. The magniude of he capial sock is defined as of he firs year of he ime sep, and he level in each subsequen ime sep is a funcion of he depreciaion rae, δ, and he invesmen during he period, i, k ) n + 1 = n i + (1 δ k (3) where n is he number of years in each period of he model s soluion. All-facor produciviy change augmens oupu a an annual rae γ. a = a ), (4) 0 (1 + γ where his shif parameer a has an iniial level a Income, Savings, and Invesmen The oupu of he economy, ne of he coss of supplying energy, becomes he income received by a represenaive global consumer. The relaionship among hese facors is defined by he following accouning ideniy: y = c + s + (pe e ), (5) where pe is he price of energy. Also, all savings are invesed, s = i. (6) ]

4 Homework #2 p. 3 Savings behavior is modeled differenly depending on he version of he model you are working wih. In he recursive-dynamic version, saving is deermined as a fixed fracion, η, of income, s = η (7) y In he forward-looking version, saving is a variable deermined in he muli-period opimizaion, explained below. 1.3 The Cos of Energy Energy use e is measured in exajoules (EJ = J). The marginal cos of energy (and herefore he price) is deermined by he depleion of naural resources. As cumulaive energy use rises, he cos of energy per EJ increases as follows where ν r μ pe = (8) r cume r is a consan represening he fixed amoun of in-ground resources, cume, cumulaive energy consumpion, evolves according o cume = n e + cume cume 0 +1, = 0 (9) μ and ν are esimaed parameers. 1.4 Carbon Accouning The carbon inensiy of energy, ε, is fixed over ime. Emissions of CO 2, denoed m, are saed in gigaons (GT = 10 9 ons) of carbon per year. They are calculaed by muliplying he carbon inensiy of energy by he amoun of energy consumed: m = ε e (10) 1.5 Parameer Values The parameer values are: n = 10 a 0 = 0.8 r = EJ α = 0.25 β = 0.65 δ = 0.05 (number of years per period) (iniial value of echnological shif facor) (energy resource base) (capial share in producion funcion) (labor share in producion funcion) (annual depreciaion rae of capial)

5 Homework #2 p. 4 ε = GT/EJ μ = 0.01 ν = 1 σ = 1.25 ρ = (carbon coefficien on energy use) (slope parameer in energy cos funcion) (exponen parameer in energy cos funcion) (elasiciy of subsiuion) ( σ 1) / σ = 0.2 (subsiuion parameer in producion funcion) θ = 0.03 λ = 0.01 γ = 0.01 (annual discoun rae) (annual growh rae of labor supply) (annual growh rae of all facor produciviy change) The iniial-year values of he variables in he model are he following: l 0 = k 0 = 130 e 0 = 600 (10 9 worker-hours) (10 12 $US) (EJ) The ime = 0 is he year 2000, and he spreadshee provided for Pars A, B, and C of he homework se is already se up in 10-year periods. 3. TASKS WITH THE MODEL You are o prepare calculaions wih wo versions of his model, and use i o predic global emissions under various assumpions. For each version, please hand in 1. A copy of each workshee where appropriae. 2. Plos of he ime pahs of consumpion, carbon emissions, and amospheric concenraions. Feel free o plo any oher variables ha you find of ineres. 3. Commens on he soluions as requesed below. Once you have se up he workshee, you may wan o explore he effec on he resuls of alernaive values of key parameers, e.g., γ, η, σ KLE. Noe ha he model may no solve if you depar far from he reference parameers specified above. Par A. Tasks wih he Recursive-Dynamic Version The Excel emplae incorporaes a myopic version of he model, and shows a projecion of global CO 2 emissions up o 2100 (column N). This version assumes ha energy use grows from is year 2000 level of 600 EJ according o an elasiciy relaionship wih GDP, y, wih a one-period lag (column F). Tha is,

6 Homework #2 p. 5 e e 1 = 0.5 = 0.5 χ [ e y ] χ [ e ] 0 y where χ = 0.25 is he elasiciy of energy use o oupu. Solve his model in separae cases for Problem A1. A higher rae of produciviy growh (γ=0.015), and Problem A2. A lower rae of labor force growh (λ=0.005). (Reurn γ o is original value of 0.01 before running his case.) Commen briefly on he way hese changes influence he oher variables in he model. Par B. A Tasks wih he Forward-Looking Version Solve he model assuming ha he level of energy use (he conrol variable) is se over all periods o maximize he presen value of welfare. In any period, he index describing welfare is W = ln(c ). This funcional form, (column K) implemens an assumpion ha he marginal uiliy of consumpion declines as consumpion rises. Thus he funcion o be maximized is 1 max PV ( W ) = ln c (11) 1+ θ where θ is he uiliy discoun rae (rae of ime preference), which is se o θ = 0.03 on an annual basis in he workshee. The model maximizes equaion (11) by choosing he pah of e. The iniial value of his variable is fixed, wih e 0 = 600 (EJ). Problem B1. Prepare a reference ( business as usual ) scenario on he assumpion here is no carbon consrain. (Make sure produciviy growh and labor force growh are rese (γ=0.01; λ=0.01). Problem B2. Prepare a conrol scenario applying an inensiy arge of he form announced by he Bush Adminisraion and some developing counries. Assume he energy inensiy of GDP (m/y) declines by 15% per decade, beginning from he inensiy in he 2000 base year. (Hin: Se up he inensiy consrain in wih P24-P33 <= Q24-Q33 wih he limi se up in he Q column.) Solver is sensiive o saring condiions here. If you ge repeaed errors, reload he emplae and sar again.) Problem B3. Consruc a curve of marginal cos, defined in erms of los (nondiscouned) consumpion, for global emissions reducions of up o 4 GC, for he periods 2010 and This may be done by solving he model for reducions below reference values (from Problem B1) of 1.0, 2.0, 3.0, and 4.0 GC and hen ploing he marginal decrease in consumpion. Commen briefly on he shapes of he curves. Remember ha

7 Homework #2 p. 6 consumpion is in unis of rillions of dollars, and so a 0.1 change in he consumpion level is acually 100 billion dollars. (Noe also ha, because of is simpliciy, his model will yield coss higher han you will see in oher analyses, and a non-zero inercep.) Par C. Sabilizaion of Amospheric CO 2 Concenraion A simple model of CO 2 accumulaion (in GC) in he amosphere is where and n skm = n m skm 1 (12) τ e skm 0 = GC (carbon in amosphere in 2000) τ e = 120 years (e-folding ime of carbon in he amosphere). The amospheric concenraion, in ppmv, is hen conc = skm / The emplae already includes wo columns for he calculaion of amospheric concenraions of CO 2, and a figure o display he resuls is provided in he workshee. Implemen hese equaions and conduc he following exercises using he forward-looking model. Problem C1. Solve for he paern of energy use (and associaed carbon emissions) ha sabilizes amospheric carbon concenraions a or below 550 ppmv for every period of he model. Wha is he yearly carbon upake in 2100 under his scenario? Commen on he model assumpion abou erresrial and ocean carbon upake. Problem C2. Solve he same case as C1, only wih he discoun rae reduced from 0.03 o Commen briefly on he difference his makes o he soluion. (Do no forge o rerun he solver, oherwise he opimizaion doesn happen and all you do is change he ne presen value.) Problem C3. Solve he C1 case again (i.e., <= 550ppmv for all periods), only assume ha because of poliical consrains he level of energy use canno be reduced more rapidly han 10% per decade. Commen briefly on how he soluions differ. Par D. How o Improve he Model Lis hree addiional feaures you would add o he model o make i more realisic, commening briefly on 1. Which you believe are he mos imporan for undersanding he climae issue? 2. Wha problems (daa, analysis mehods) would arise in implemenaion? 3. Are here issues ha are imporan ha his ype of model would never be able o address, even wih significanly more complexiy?

8 Homework #2 p SETUP OF THE SAMPLE WORKSHEET TEMPLATE The problem is se up wih 11 en-year periods, * = 0,...,10. For each period, * he value of he economic variables (y, k, l, c, i) are se a he appropriae annual levels of he firs year in he 10-year calculaion sep. Tha is, y(* = 0) y( = 0) y(* = 1) y( = 10) ec. Thus equaions (3), (9) and (12) assume ha in he firs year of he 10-year ime sep invesmen, energy and emissions (respecively) apply for all years of ha sep, so ha hese (annual) quaniies mus be muliplied by n = 10. Similarly, in equaions (3) and (12) he annual survival facors (1 - δ) and (1-1/τ e ) mus be updaed over an enire decade, which is accomplished by raising hem o he power n. 5. WORKSHEET HINTS 1. Before beginning work save a maser copy of he workshee (o be lef unchanged) and se up a working version. You may need o exi and sar over and his will save you having o go back o he web. 2. This noe is wrien as if he dynamic version of he model is o be solved in Excel using Solver. Equivalen faciliies should be available in oher spreadshee programs. The policies limiing emissions and/or concenraions mus be enered as consrains in Solver. 3. When solving he dynamic model, bounds someimes mus be placed on values of he variables o ensure ha Solver finds an opimum ha in fac makes economic sense. If you run ino problems ry ensuring ha energy use, energy prices, consumpion and invesmen canno be negaive. Rule ou hese possibiliies by adding he consrains e 0, pe 0, c 0 and i 0 for all. 4. When solving he dynamic model under consrain, always re-run he unconsrained version before running wih a differen consrain. Solver does no always move accuraely from one consrained soluion o anoher. Also, someimes Solver ges suck (e.g., when confroned wih a very resricive consrain), and canno reurn o he noconsrain soluion. Therefore, you should always run an unconsrained version of any problem before moving from one consrained soluion o anoher. 5. Someimes Solver reurns a soluion ha is no an opimum. If you run ino his problem, ofen you can recover by limiing he zone of search for he opimal soluion and performing successive solves, using he soluion of one as he iniial poin for he nex. This process should converge o a sensible answer.

Macroeconomic 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 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 information

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,

More information

Economics 8105 Macroeconomic Theory Recitation 6

Economics 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 information

Cooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.

Cooperative 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 information

3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon

3.1.3 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 information

Suggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class

Suggested 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 information

Final Exam. Tuesday, December hours

Final 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 information

The general Solow model

The general Solow model The general Solow model Back o a closed economy In he basic Solow model: no growh in GDP per worker in seady sae This conradics he empirics for he Wesern world (sylized fac #5) In he general Solow model:

More information

Examples of Dynamic Programming Problems

Examples 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 information

Introduction to choice over time

Introduction to choice over time Microeconomic Theory -- Choice over ime Inroducion o choice over ime Individual choice Income and subsiuion effecs 7 Walrasian equilibrium ineres rae 9 pages John Riley Ocober 9, 08 Microeconomic Theory

More information

( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor

( ) (, ) 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 information

T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 2011 EXAMINATION

T. 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

Seminar 4: Hotelling 2

Seminar 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 information

Essential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems

Essential Microeconomics : OPTIMAL CONTROL 1. Consider the following class of optimization problems Essenial Microeconomics -- 6.5: OPIMAL CONROL Consider he following class of opimizaion problems Max{ U( k, x) + U+ ( k+ ) k+ k F( k, x)}. { x, k+ } = In he language of conrol heory, he vecor k is he vecor

More information

Lecture Notes 5: Investment

Lecture Notes 5: Investment Lecure Noes 5: Invesmen Zhiwei Xu (xuzhiwei@sju.edu.cn) Invesmen decisions made by rms are one of he mos imporan behaviors in he economy. As he invesmen deermines how he capials accumulae along he ime,

More information

Final Exam Advanced Macroeconomics I

Final Exam Advanced Macroeconomics I Advanced Macroeconomics I WS 00/ Final Exam Advanced Macroeconomics I February 8, 0 Quesion (5%) An economy produces oupu according o α α Y = K (AL) of which a fracion s is invesed. echnology A is exogenous

More information

Final Spring 2007

Final 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 information

This document was generated at 7:34 PM, 07/27/09 Copyright 2009 Richard T. Woodward

This 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 information

6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010.

6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010. 6.003 Homework Due a he beginning of reciaion on Wednesday, February 0, 200. Problems. Independen and Dependen Variables Assume ha he heigh of a waer wave is given by g(x v) where x is disance, v is velociy,

More information

Solutions Problem Set 3 Macro II (14.452)

Solutions 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 information

Explaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015

Explaining 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 information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

Appendix 14.1 The optimal control problem and its solution using

Appendix 14.1 The optimal control problem and its solution using 1 Appendix 14.1 he opimal conrol problem and is soluion using he maximum principle NOE: Many occurrences of f, x, u, and in his file (in equaions or as whole words in ex) are purposefully in bold in order

More information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 6 SECTION 6.1: LIFE CYCLE CONSUMPTION AND WEALTH T 1. . Let ct. ) is a strictly concave function of c John Riley December 00 S O EVEN NUMBERED EXERCISES IN CHAPER 6 SECION 6: LIFE CYCLE CONSUMPION AND WEALH Eercise 6-: Opimal saving wih more han one commodiy A consumer has a period uiliy funcion δ u (

More information

Problem 1 / 25 Problem 2 / 20 Problem 3 / 10 Problem 4 / 15 Problem 5 / 30 TOTAL / 100

Problem 1 / 25 Problem 2 / 20 Problem 3 / 10 Problem 4 / 15 Problem 5 / 30 TOTAL / 100 eparmen of Applied Economics Johns Hopkins Universiy Economics 602 Macroeconomic Theory and Policy Miderm Exam Suggesed Soluions Professor Sanjay hugh Fall 2008 NAME: The Exam has a oal of five (5) problems

More information

A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18

A 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 information

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.

More information

Math 333 Problem Set #2 Solution 14 February 2003

Math 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 information

Solutions to Assignment 1

Solutions 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 information

Full file at

Full file at Full file a hps://frasockeu SOLUTIONS TO CHAPTER 2 Problem 2 (a) The firm's problem is o choose he quaniies of capial, K, and effecive labor, AL, in order o minimize coss, wal + rk, subjec o he producion

More information

E 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.

E 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 information

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Chapter 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 information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION DOI: 0.038/NCLIMATE893 Temporal resoluion and DICE * Supplemenal Informaion Alex L. Maren and Sephen C. Newbold Naional Cener for Environmenal Economics, US Environmenal Proecion

More information

Problem Set #1 - Answers

Problem Set #1 - Answers Fall Term 24 Page of 7. Use indifference curves and a curved ransformaion curve o illusrae a free rade equilibrium for a counry facing an exogenous inernaional price. Then show wha happens if ha exogenous

More information

KINEMATICS IN ONE DIMENSION

KINEMATICS 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 information

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3 and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or

More information

Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 1

Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 1 Macroeconomics I, UPF Professor Anonio Ciccone SOUTIONS PROBEM SET. (from Romer Advanced Macroeconomics Chaper ) Basic properies of growh raes which will be used over and over again. Use he fac ha he growh

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON4325 Moneary Policy Dae of exam: Tuesday, May 24, 206 Grades are given: June 4, 206 Time for exam: 2.30 p.m. 5.30 p.m. The problem se covers 5 pages

More information

Lecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model

Lecture 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 information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. 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 information

Problem set 3: Endogenous Innovation - Solutions

Problem set 3: Endogenous Innovation - Solutions Problem se 3: Endogenous Innovaion - Soluions Loïc Baé Ocober 25, 22 Opimaliy in he R & D based endogenous growh model Imporan feaure of his model: he monopoly markup is exogenous, so ha here is no need

More information

Online Appendix to Solution Methods for Models with Rare Disasters

Online Appendix to Solution Methods for Models with Rare Disasters Online Appendix o Soluion Mehods for Models wih Rare Disasers Jesús Fernández-Villaverde and Oren Levinal In his Online Appendix, we presen he Euler condiions of he model, we develop he pricing Calvo block,

More information

Errata (1 st Edition)

Errata (1 st Edition) P Sandborn, os Analysis of Elecronic Sysems, s Ediion, orld Scienific, Singapore, 03 Erraa ( s Ediion) S K 05D Page 8 Equaion (7) should be, E 05D E Nu e S K he L appearing in he equaion in he book does

More information

Lecture 3: Solow Model II Handout

Lecture 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 information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

The Brock-Mirman Stochastic Growth Model

The Brock-Mirman Stochastic Growth Model c December 3, 208, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner

More information

= ( ) ) or a system of differential equations with continuous parametrization (T = R

= ( ) ) or a system of differential equations with continuous parametrization (T = R XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

Problem Set #3: AK models

Problem 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 information

The average rate of change between two points on a function is d t

The 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 information

Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories

More information

Reserves measures have an economic component eg. what could be extracted at current prices?

Reserves 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 information

A Dynamic Model of Economic Fluctuations

A 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 information

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively: XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

Chapter 2. First Order Scalar Equations

Chapter 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

A Note on Raising the Mandatory Retirement Age and. Its Effect on Long-run Income and Pay As You Go (PAYG) Pensions

A Note on Raising the Mandatory Retirement Age and. Its Effect on Long-run Income and Pay As You Go (PAYG) Pensions The Sociey for Economic Sudies The Universiy of Kiakyushu Working Paper Series No.2017-5 (acceped in March, 2018) A Noe on Raising he Mandaory Reiremen Age and Is Effec on Long-run Income and Pay As You

More information

Vehicle Arrival Models : Headway

Vehicle 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 information

Matlab and Python programming: how to get started

Matlab and Python programming: how to get started Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,

More information

Lab #2: Kinematics in 1-Dimension

Lab #2: Kinematics in 1-Dimension Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion

More information

Decomposing Value Added Growth Over Sectors into Explanatory Factors

Decomposing Value Added Growth Over Sectors into Explanatory Factors Business School Decomposing Value Added Growh Over Secors ino Explanaory Facors W. Erwin Diewer (UBC and UNSW Ausralia) and Kevin J. Fox (UNSW Ausralia) EMG Workshop UNSW 2 December 2016 Summary Decompose

More information

IMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013

IMPLICIT AND INVERSE FUNCTION THEOREMS PAUL SCHRIMPF 1 OCTOBER 25, 2013 IMPLICI AND INVERSE FUNCION HEOREMS PAUL SCHRIMPF 1 OCOBER 25, 213 UNIVERSIY OF BRIISH COLUMBIA ECONOMICS 526 We have exensively sudied how o solve sysems of linear equaions. We know how o check wheher

More information

04. Kinetics of a second order reaction

04. Kinetics of a second order reaction 4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius

More information

Solutions to Odd Number Exercises in Chapter 6

Solutions to Odd Number Exercises in Chapter 6 1 Soluions o Odd Number Exercises in 6.1 R y eˆ 1.7151 y 6.3 From eˆ ( T K) ˆ R 1 1 SST SST SST (1 R ) 55.36(1.7911) we have, ˆ 6.414 T K ( ) 6.5 y ye ye y e 1 1 Consider he erms e and xe b b x e y e b

More information

GMM - Generalized Method of Moments

GMM - Generalized Method of Moments GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................

More information

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates

Biol. 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 information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

The fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation

The fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or

More information

6.003 Homework #9 Solutions

6.003 Homework #9 Solutions 6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j

More information

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

1 Consumption and Risky Assets

1 Consumption and Risky Assets Soluions o Problem Se 8 Econ 0A - nd Half - Fall 011 Prof David Romer, GSI: Vicoria Vanasco 1 Consumpion and Risky Asses Consumer's lifeime uiliy: U = u(c 1 )+E[u(c )] Income: Y 1 = Ȳ cerain and Y F (

More information

SZG Macro 2011 Lecture 3: Dynamic Programming. SZG macro 2011 lecture 3 1

SZG Macro 2011 Lecture 3: Dynamic Programming. SZG macro 2011 lecture 3 1 SZG Macro 2011 Lecure 3: Dynamic Programming SZG macro 2011 lecure 3 1 Background Our previous discussion of opimal consumpion over ime and of opimal capial accumulaion sugges sudying he general decision

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 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 information

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure Lab 4: Synchronous Sae Machine Design Summary: Design and implemen synchronous sae machine circuis and es hem wih simulaions in Cadence Viruoso. Learning Objecives: Pracice designing and simulaing digial

More information

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models

More information

5.2. The Natural Logarithm. Solution

5.2. The Natural Logarithm. Solution 5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,

More information

Seminar 5 Sustainability

Seminar 5 Sustainability Seminar 5 Susainabiliy Soluions Quesion : Hyperbolic Discouning -. Suppose a faher inheris a family forune of 0 million NOK an he wans o use some of i for himself (o be precise, he share ) bu also o beques

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On 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 information

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling?

not 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 information

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x . 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih

More information

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H. ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models

More information

BU Macro BU Macro Fall 2008, Lecture 4

BU Macro BU Macro Fall 2008, Lecture 4 Dynamic Programming BU Macro 2008 Lecure 4 1 Ouline 1. Cerainy opimizaion problem used o illusrae: a. Resricions on exogenous variables b. Value funcion c. Policy funcion d. The Bellman equaion and an

More information

5.1 - Logarithms and Their Properties

5.1 - 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 information

Two Coupled Oscillators / Normal Modes

Two 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 information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 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 information

10. State Space Methods

10. State Space Methods . Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he

More information

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17 EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive

More information

) were both constant and we brought them from under the integral.

) 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 information

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.

Introduction 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 information

Morning Time: 1 hour 30 minutes Additional materials (enclosed):

Morning Time: 1 hour 30 minutes Additional materials (enclosed): ADVANCED GCE 78/0 MATHEMATICS (MEI) Differenial Equaions THURSDAY JANUARY 008 Morning Time: hour 30 minues Addiional maerials (enclosed): None Addiional maerials (required): Answer Bookle (8 pages) Graph

More information

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS Name SOLUTIONS Financial Economerics Jeffrey R. Russell Miderm Winer 009 SOLUTIONS You have 80 minues o complee he exam. Use can use a calculaor and noes. Try o fi all your work in he space provided. If

More information

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions

Inventory 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 information

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

dt = 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 information

A Note on Public Debt, Tax-Exempt Bonds, and Ponzi Games

A Note on Public Debt, Tax-Exempt Bonds, and Ponzi Games WP/07/162 A Noe on Public Deb, Tax-Exemp Bonds, and Ponzi Games Berhold U Wigger 2007 Inernaional Moneary Fund WP/07/162 IMF Working Paper Fiscal Affairs Deparmen A Noe on Public Deb, Tax-Exemp Bonds,

More information

Sub Module 2.6. Measurement of transient temperature

Sub Module 2.6. Measurement of transient temperature Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,

More information

6.01: Introduction to EECS I Lecture 8 March 29, 2011

6.01: Introduction to EECS I Lecture 8 March 29, 2011 6.01: Inroducion o EES I Lecure 8 March 29, 2011 6.01: Inroducion o EES I Op-Amps Las Time: The ircui Absracion ircuis represen sysems as connecions of elemens hrough which currens (hrough variables) flow

More information

Sterilization D Values

Sterilization 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 information

STATE-SPACE MODELLING. A mass balance across the tank gives:

STATE-SPACE MODELLING. A mass balance across the tank gives: B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing

More information

COMPETITIVE GROWTH MODEL

COMPETITIVE GROWTH MODEL COMPETITIVE GROWTH MODEL I Assumpions We are going o now solve he compeiive version of he opimal growh moel. Alhough he allocaions are he same as in he social planning problem, i will be useful o compare

More information

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

Section 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous

More information

The Simple Linear Regression Model: Reporting the Results and Choosing the Functional Form

The Simple Linear Regression Model: Reporting the Results and Choosing the Functional Form Chaper 6 The Simple Linear Regression Model: Reporing he Resuls and Choosing he Funcional Form To complee he analysis of he simple linear regression model, in his chaper we will consider how o measure

More information