The Optimal Timing of Transition to New Environmental Technology in Economic Growth

Size: px
Start display at page:

Download "The Optimal Timing of Transition to New Environmental Technology in Economic Growth"

Transcription

1 h Opimal iming of ransiion o Nw Environmnal chnology in Economic Growh h IAEE Europan Confrnc 7- Spmbr, 29 Vinna, Ausria Akira AEDA and akiko NAGAYA yoo Univrsiy Background: Growh and h Environmn Naural rsourc us along w/ conomic growh has bn sudid sinc h 97s. Dasgpa and Hal (974), Sigliz (974), Solow (974), c. In h 98s, h nvironmn bgan o b rcognizd as anohr imporan facor ha drmins h rajcoris of growh. A h sam im, h hory of conomic growh bgan o chang. ndognous chang coms o a cnral issu in h conomic liraur, inc. Romr (99)Lucas (988), c. hs wo srams convrg o a collcion of sudis afr h la 99s. Endognous chnological chang plays cnral rols in susainabl dvlopmn. Barbir (999), ahvonn and Salo (2), Bovnbrg and Smuldrs (995), Schou (2), Cunha--Sá and Ris (27) 2

2 Background: Opimal iming () Cunha--Sá and Ris (27). h Opimal iming of Adopion of a Grn chnology. Environmnal and Rsourc Economics 36. hir focus: As h conomy grows, nw polluion-abaing chnology bcoms indispnsabl; Such a chnology rquirs invsmn ffors a h im of dploymn; h opimal iming will b drmind in an ndognous mannr. hir modl is basd on a ypical Ramsy-yp modl, incorporaing nvironmnal qualiy and clan chnology. h lvl of clan chnology may chang disconinuously onc upgrad happns. Considraion for h n bnfi drmins h opimal iming of upgrad. 3 Background: Opimal iming (2) Cunha--Sá and Ris conribud o h liraur in ha: hy undrlind h significanc of h opimal iming in h framwork of h nvironmn and conomic growh. Som dbaabl issus rmain on h ohr hand: chnological chang in clan chnology is rprsnd as a jump of h lvl. Allowing such a suddn chang of h lvl implis ha h chnology is a kind of flow, no sock. his naur conrass o h fundamnal ida of modrn ndognous growh hory: hir ramn of chnological chang for clan chnology hus sms old-fashiond in his rspc. hir ramn of invsmn-rlad coss sms srang. I may dgrad h clar cu of hir analysis. 4

3 h Purpos is o xamin h choic of iming for chnological chang vis-à-vis nvironmnal qualiy in conomic dvlopmn. Basd on h spiri and mahmaical ramn of Cunha--Sá and Ris, w dvlop an analyical modl ha addrsss h opimal iming of ransiion of nvironmnal chnology from h old on o h nw on. In conras o Cunha--Sá and Ris, w focus upon acclraion of chnological progrss, and w ra coss in a simplr mannr so ha our focus can bcom clarr. 5 Analyical Fram () Ramsy modl w/ closd conomy; consan populaion Y: Producion; : Capial sock; C: Consumpion Q: Environmnal qualiy a: Environmnal chnology lvl Y A d d A C A rprsnaiv conomic agn rprsns h houshold whos im-addiiv uiliy is drmind by no only consumpion (C) bu also nvironmnal qualiy (Q). µ ( C Q ) U < µ <, >, µ ( ) <, µ ( ) < 6

4 Analyical Fram (2) h nvironmnal qualiy (Q) is a funcion of h consumpion (C) and nvironmnal chnology lvl (a) Q ac > h progrss of h nvironmnal chnology lvl islf may occur as h conomy grows. h growh ras of a and ( Y/A) ar proporional o ach ohr: ( da d) a ( d d) Inroducing a posiiv cofficin η : da d d d a h cofficin η may chang disconinuously a im. 7 Analyical Fram (3) Disconinuous chang in h cofficin η: da d d d a $ for < $ for lim, < Coss for h chang ypical coss: R&D and physical invsmns Exising physical capial may bcom obsol, and nd o b scrappd. < lim, < 8

5 h odl J ( ) s.. ax U d { C } $ +, $ µ ( C Q ) U U < µ <, >, d d A C < Q a C a $ a a % % for for > d 9 Solving h Problm Conrol variabls:, { C } [, ] Sa variabls: { } [, ] Firs sp: Opimizing for h valu funcion o a givn capial sock for afr-ransiion. $ ( ) %( ) ax U d { C } Scond sp: Finding h opimal pah unil h ransiion. V ( ) ax { C } $ U hird sp: Solving for h opimal iming. J ( ) axv( ) d + ( )

6 Firs Sp Suppos ha ransiion o h nw chnology occurrd a im. h problm hrafr is: & ( ) s.. ax { C } % µ % ( CQ ) $ ( ) d d d A C Q a C a a FONCs: µ % & µ % $µ % H C $µ a C C ( ) ( ) ( ) ( )( ) µ ( $ ) µ ( & $ ) ( & %µ )( & $ ) ' H ' µ a C % lim $ & & d + A & d Soluion for h Firs Sp C > whr dc d d d C $ ) µ + % (µ > % (µ * % A & ( % ' )( ) µ + % (µ) %( % (µ)( % ' ) (consan) A µ + ( % µ)( A % $ ) [ % ( % µ)( % )]( µ + % µ) > W obain: ( ) a µ ( & )-' +, ( & $µ )( & ) * ( & )( % µ + & $µ ) ( ) & & ' ( ) ( & ) 2

7 Scond Sp Finding h opimal pah o im V FONCs: ( ) s.. ax { C } % d d A C $ ac µ $ ( C Q ) $ $ $ ( ) < Q a a $ d + µ ( % ) & µ ( % ) ( $µ )( % ) ( $µ ) a C H C C µ ( $ ) µ ( & $ ) ( & %µ )( & $ ) ' H ' µ a C $( ) & & d + A & d % 3 Soluion for h Scond Sp C + whr ( ) ( ) $ ) µ + % (µ > % (µ & & ( $ µ )( $ ) C $ C $ $ ( $ µ )( $ ) $ * % A( % ' )( ) µ + % (µ) > % ( % (µ)( % ' ) ( $ µ + ' µ){% ' A( ' )( $ µ + ' µ)} ( $ µ + ' µ){% ' A( ' )( $ µ + ' µ)} ( ' µ )( ' ) ' ( ' µ )( ' ) W obain: A & ( $ ) ' % (* ' ) ) (* ' ) ') ( ' ) + o ') ( + 4

8 h Final Sp Opimizing i w.r.. J( ) axv( ) whr V ( ) ( ) ax U d + { C } $ has alrady bn obaind analyically. hus, w obain: ( )( ) ( ( ( 2µ ( µ ) ( ( 2µ )( ( ) dv ( ) a '. / + $ ( )( ) ' + ( )( ) ( )( ) ( ( ) $ ( 3 ( µ + ( 2µ &, ) & ( ( ( µ ( ( 2µ ( µ ( ( 4 d ( % - * 5 % Examining is sign, w know h propris of h soluion. dv ( ) d * if < * < dv ( ) d * < if * 5 Proposiion For a fini * o xis, ha is, * <, h following condiion is ncssary: Convrsly, if ( $ )( A ) or, quivalnly ( )( A ) < > ' and A & $ < ' and A % $ holds, hn h ransiion of nvironmnal chnology nvr occurs. (i.. * ) 6

9 Inrpraion of Prop. () h xisnc of a fini ransiion im is drmind by: Noic: '( $ ()% A & * + ' $ C A C Y % < & < his rprsns avrag consumpion propnsiy jus bfor h ransiion. σ rprsns h magniud of h lasiciy of marginal uiliy. > ' and A & $ < ' and A % $ I drmins h sign of h cross scond drivaiv of h uiliy 2 funcion: U { µ ( ) } µ ( ) C Q C Q If >σ (<σ), marginal uiliy of nvironmnal qualiy U/ Q is incrasing (dcrasing) in consumpion. ha is, h sociy prfrs mor (lss) nvironmnal qualiy as h consumpion grows. 7 Inrpraion of Prop. (2) For h ransiion o occur in a fini im horizon, h conomy mus b in ihr of h following wo siuaions: h conomy highly prfrs nvironmnal qualiy, and is currn avrag consumpion propnsiy is lowr han a crain valu (/δ). (h conomy has a highr saving ra.) h conomy rahr prfrs consumpion o nvironmnal qualiy, and is currn avrag consumpion propnsiy is highr han a crain valu (/δ). (h conomy has a lowr saving ra.) 8

10 Proposiion 2 Suppos ha h following condiions hold. ( )( A ) > And, ihr or + % * > and < ( $ ) + % * < and > ( $ ) ( % µ ) ( + µ ' {( % ) µ } %( % µ )( % ) & ( % µ ) ( + µ ' {( % ) µ } %( % µ )( % ) & hn hr xiss an opimal ransiion im * such ha < *<, and i is obaind by solving h following quaions: $,, % [ + $ ] o A h capial sock a h im: + % * ( $ ) ( % µ ) ( % µ )( % ) ( + µ ' % % & {( ) µ } 9 Inrpraion of Prop. 2 Givn h xisnc of a fini *, > cas: Capial sock incrass, and rachs h fixd lvl from blow. < cas: Capial sock dclins, and rachs h fixd lvl from abov. > < 2

11 Proposiion 3 ( )( ) Suppos A > Also, suppos h following approximaions hold. µ hn, w hav h following snsiiviis. d d ( ) < > cas: < cas: d d < ( ) d d > ( ) No: da d d d a $ $ for for < 2 Inrpraion of Prop. 3 rprsns h dgr of h innovaion. Inuiivly, i lls how much h ransiion o h nw nvironmnal chnology is bnficial o h conomy. h incras maks h barrir o ransiion lowr. > < 22

12 Conclusion W invsigad h opimal iming of ransiion in nvironmnal chnology from h old on o h nw on along wih conomic growh. Whhr ransiion o nw chnology is ndd or no is drmind by wo facors: h dgr of complmnariy bwn nvironmnal qualiy and consumpion, and h avrag consumpion propnsiy (or saving ra). Givn h ncssiy, ransiions o nw chnology may occur in wo possibl ways. An nvironmnally dvloping conomy would accumula social capial and invs i o h dvlopmn of h nw chnology. A maurd conomy holding a sufficincy of capial and njoying a high consumpion ra would raliz h nd for nvironmnal qualiy improvmn a som im. A highr dgr of innovaion would lad o arlir or lar ransiions, rspcivly. 23

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

More information

Midterm Examination (100 pts)

Midterm Examination (100 pts) Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion

More information

Mundell-Fleming I: Setup

Mundell-Fleming I: Setup Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)

More information

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT [Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI

More information

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison Economics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 3/28/2012 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 16 1 Consumpion Th Vry Forsighd dconsumr A vry forsighd

More information

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions 4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway

More information

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions

More information

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t

5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:

1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to: Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding

More information

Microscopic Flow Characteristics Time Headway - Distribution

Microscopic Flow Characteristics Time Headway - Distribution CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,

More information

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...

More information

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER

A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:

More information

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -

More information

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form:

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form: Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,

More information

EXERCISE - 01 CHECK YOUR GRASP

EXERCISE - 01 CHECK YOUR GRASP DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc

More information

Charging of capacitor through inductor and resistor

Charging of capacitor through inductor and resistor cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 4/25/2011. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 4/25/2011. UW Madison conomics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 4/25/2011 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 21 1 Th Mdium Run ε = P * P Thr ar wo ways in which

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

Control System Engineering (EE301T) Assignment: 2

Control System Engineering (EE301T) Assignment: 2 Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also

More information

The Overlapping Generations growth model. of Blanchard and Weil

The Overlapping Generations growth model. of Blanchard and Weil 1 / 35 Th Ovrlapping Gnraions growh modl of Blanchard and Wil Novmbr 15, 2015 Alcos Papadopoulos PhD Candida Dparmn of Economics Ahns Univrsiy of Economics and Businss papadopalx@aub.gr I prsn a daild

More information

THE SHORT-RUN AGGREGATE SUPPLY CURVE WITH A POSITIVE SLOPE. Based on EXPECTATIONS: Lecture. t t t t

THE SHORT-RUN AGGREGATE SUPPLY CURVE WITH A POSITIVE SLOPE. Based on EXPECTATIONS: Lecture. t t t t THE SHORT-RUN AGGREGATE SUL CURVE WITH A OSITIVE SLOE. Basd on EXECTATIONS: Lcur., 0. In Mankiw:, 0 Ths quaions sa ha oupu dvias from is naural ra whn h pric lvl dvias from h xpcd pric lvl. Th paramr a

More information

I) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning

I) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic

More information

Lecture 4: Laplace Transforms

Lecture 4: Laplace Transforms Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions

More information

Logistic equation of Human population growth (generalization to the case of reactive environment).

Logistic equation of Human population growth (generalization to the case of reactive environment). Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru

More information

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs

More information

Solutions to End-of-Chapter Problems for Chapters 26 & 27 in Textbook

Solutions to End-of-Chapter Problems for Chapters 26 & 27 in Textbook Soluions o End-of-Chapr Problms for Chaprs 26 & 27 in Txbook Chapr 26. Answrs o hs Tru/Fals/Uncrain can b found in h wrin x of Chapr 26. I is lf o h sudn o drmin h soluions. 2. For his qusion kp in mind

More information

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th

More information

14.02 Principles of Macroeconomics Problem Set 5 Fall 2005

14.02 Principles of Macroeconomics Problem Set 5 Fall 2005 40 Principls of Macroconomics Problm S 5 Fall 005 Posd: Wdnsday, Novmbr 6, 005 Du: Wdnsday, Novmbr 3, 005 Plas wri your nam AND your TA s nam on your problm s Thanks! Exrcis I Tru/Fals? Explain Dpnding

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

Institute of Actuaries of India

Institute of Actuaries of India Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6

More information

Lecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey

Lecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd

More information

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is 39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h

More information

Methodology for Analyzing State Tax Policy By Orphe Pierre Divounguy, PhD, Revised by Andrew J. Kidd, PhD (May 2018)

Methodology for Analyzing State Tax Policy By Orphe Pierre Divounguy, PhD, Revised by Andrew J. Kidd, PhD (May 2018) Mhodology for Analyzing Sa Tax Policy By Orph Pirr Divounguy, PhD, Rvisd by Andrw J. Kidd, PhD (May 2018) Inroducion To analyz how changs o ax policy impacs no only govrnmn rvnus bu also conomic aciviy

More information

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +

More information

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )

AR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 ) AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc

More information

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate A Condiion for abiliy in an I Ag rucurd Disas Modl wih Dcrasing urvival a A.K. upriana, Edy owono Dparmn of Mahmaics, Univrsias Padjadjaran, km Bandung-umng 45363, Indonsia fax: 6--7794696, mail: asupria@yahoo.com.au;

More information

UNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED

UNSTEADY FLOW OF A FLUID PARTICLE SUSPENSION BETWEEN TWO PARALLEL PLATES SUDDENLY SET IN MOTION WITH SAME SPEED 006-0 Asian Rsarch Publishing work (ARP). All righs rsrvd. USTEADY FLOW OF A FLUID PARTICLE SUSPESIO BETWEE TWO PARALLEL PLATES SUDDELY SET I MOTIO WITH SAME SPEED M. suniha, B. Shankr and G. Shanha 3

More information

Impulsive Differential Equations. by using the Euler Method

Impulsive Differential Equations. by using the Euler Method Applid Mahmaical Scincs Vol. 4 1 no. 65 19 - Impulsiv Diffrnial Equaions by using h Eulr Mhod Nor Shamsidah B Amir Hamzah 1 Musafa bin Mama J. Kaviumar L Siaw Chong 4 and Noor ani B Ahmad 5 1 5 Dparmn

More information

Final Exam : Solutions

Final Exam : Solutions Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

Lecture 2: Current in RC circuit D.K.Pandey

Lecture 2: Current in RC circuit D.K.Pandey Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging

More information

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

More information

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract

General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields! Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr

More information

The Firm s Value and Timing of Adopting E-Commerce Using Real Options Analysis under Uncertainty *

The Firm s Value and Timing of Adopting E-Commerce Using Real Options Analysis under Uncertainty * Th Firm s Valu and Timing of Adoping E-Commrc Using Ral Opions Analysis undr Uncrainy * David S. Shyu Dparmn of Financ, Naional Sun Ya-sn Univrsiy, Kaohsiung, Taiwan dshyu@cm.nsysu.du.w Kuo-Jung L Dparmn

More information

Lagrangian for RLC circuits using analogy with the classical mechanics concepts

Lagrangian for RLC circuits using analogy with the classical mechanics concepts Lagrangian for RLC circuis using analogy wih h classical mchanics concps Albrus Hariwangsa Panuluh and Asan Damanik Dparmn of Physics Educaion, Sanaa Dharma Univrsiy Kampus III USD Paingan, Maguwoharjo,

More information

H is equal to the surface current J S

H is equal to the surface current J S Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

Poisson process Markov process

Poisson process Markov process E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc

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

Journal of Applied Science and Agriculture

Journal of Applied Science and Agriculture Journal of Applid Scinc and Agriculur, 93 March 2014, Pags: 1066-1070 AENS Journals Journal of Applid Scinc and Agriculur SSN 1816-9112 Journal hom pag: www.ansiwb.com/jasa/indx.hml h Opimal ax Ra in Middl

More information

Problem Set on Differential Equations

Problem Set on Differential Equations Problem Se on Differenial Equaions 1. Solve he following differenial equaions (a) x () = e x (), x () = 3/ 4. (b) x () = e x (), x (1) =. (c) xe () = + (1 x ()) e, x () =.. (An asse marke model). Le p()

More information

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if. Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[

More information

EE 434 Lecture 22. Bipolar Device Models

EE 434 Lecture 22. Bipolar Device Models EE 434 Lcur 22 Bipolar Dvic Modls Quiz 14 Th collcor currn of a BJT was masurd o b 20mA and h bas currn masurd o b 0.1mA. Wha is h fficincy of injcion of lcrons coming from h mir o h collcor? 1 And h numbr

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

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

On the Speed of Heat Wave. Mihály Makai

On the Speed of Heat Wave. Mihály Makai On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.

More information

3(8 ) (8 x x ) 3x x (8 )

3(8 ) (8 x x ) 3x x (8 ) Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6

More information

William Barnett. Abstract

William Barnett. Abstract Inrmporally non sparabl monary ass risk adjusmn aggrgaion William Barn Univrsiy o ansas Shu Wu Univrsiy o ansas Absrac Modrn aggrgaion hory indx numbr hory wr inroducd ino monary aggrgaion by Barn (980.

More information

XV Exponential and Logarithmic Functions

XV Exponential and Logarithmic Functions MATHEMATICS 0-0-RE Dirnial Calculus Marin Huard Winr 08 XV Eponnial and Logarihmic Funcions. Skch h graph o h givn uncions and sa h domain and rang. d) ) ) log. Whn Sarah was born, hr parns placd $000

More information

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy

More information

A HAMILTON-JACOBI TREATMENT OF DISSIPATIVE SYSTEMS

A HAMILTON-JACOBI TREATMENT OF DISSIPATIVE SYSTEMS Europan Scinific Journal Ocobr 13 diion vol9, No3 ISSN: 1857 7881 (Prin) - ISSN 1857-7431 A AMILTON-JACOBI TREATMENT OF DISSIPATIVE SYSTEMS Ola A Jarab'ah Tafila Tchnical Univrsiy, Tafila, Jordan Khald

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

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

The Brock-Mirman Stochastic Growth Model

The Brock-Mirman Stochastic Growth Model c November 20, 207, 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

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas

whereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically

More information

Relation between Fourier Series and Transform

Relation between Fourier Series and Transform EE 37-3 8 Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Rlion bwn ourir Sris n Trnsform Th ourir Trnsform T is riv from h finiion of h ourir Sris S. Consir, for xmpl, h prioic complx sinl To wih prio

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

Smoking Tobacco Experiencing with Induced Death

Smoking Tobacco Experiencing with Induced Death Europan Journal of Biological Scincs 9 (1): 52-57, 2017 ISSN 2079-2085 IDOSI Publicaions, 2017 DOI: 10.5829/idosi.jbs.2017.52.57 Smoking Tobacco Exprincing wih Inducd Dah Gachw Abiy Salilw Dparmn of Mahmaics,

More information

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)

46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r) 98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +

More information

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi

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

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm

More information

PAYG pensions and economic cycles

PAYG pensions and economic cycles MPRA Munich Prsonal RPEc Archiv PAYG pnsions and conomic cycls Luciano Fani and Luca Gori Univrsiy of Pisa, Univrsiy of Pisa January 00 Onlin a hps://mpra.ub.uni-munchn.d/9984/ MPRA Papr No. 9984, posd

More information

PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY. T. C. Koopmans

PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY. T. C. Koopmans PROOF FOR A CASE WHERE DISCOUNTING ADVANCES THE DOOMSDAY T. C. Koopmans January 1974 WP-74-6 Working Papers are no inended for disribuion ouside of IIASA, and are solely for discussion and informaion purposes.

More information

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )

Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( ) Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division

More information

Economic Growth & Development: Part 4 Vertical Innovation Models. By Kiminori Matsuyama. Updated on , 11:01:54 AM

Economic Growth & Development: Part 4 Vertical Innovation Models. By Kiminori Matsuyama. Updated on , 11:01:54 AM Economic Growh & Developmen: Par 4 Verical Innovaion Models By Kiminori Masuyama Updaed on 20-04-4 :0:54 AM Page of 7 Inroducion In he previous models R&D develops producs ha are new ie imperfec subsiues

More information

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs

More information

International and Development Economics

International and Development Economics Crawford School of Economics and Govrnmn WORKING PAPERS H AUSTRALIAN NATIONAL UNIVERSITY Inrnaional and Dvlopmn Economics O upu vrsus Inpu Conrols undr Uncrainy: Th Cas of a Fishry 07-06 Saoshi Yamazaki

More information

Natural Resource Economics

Natural Resource Economics Naura Rsourc Economics Acamic ar: 2018-2019 Prof. Luca Savaici uca.savaici@uniroma3.i Lsson 14: Opima conro sufficin coniions Naura Rsourc Economics - Luca Savaici 2018-19 1 FOCs Saic probm: Dnamic probm:

More information

Physics 160 Lecture 3. R. Johnson April 6, 2015

Physics 160 Lecture 3. R. Johnson April 6, 2015 Physics 6 Lcur 3 R. Johnson April 6, 5 RC Circui (Low-Pass Filr This is h sam RC circui w lookd a arlir h im doma, bu hr w ar rsd h frquncy rspons. So w pu a s wav sad of a sp funcion. whr R C RC Complx

More information

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin

More information

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve

10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve 0. If p and q ar h lnghs of h prpndiculars from h origin on h angn and h normal o h curv + Mahmaics y = a, hn 4p + q = a a (C) a (D) 5a 6. Wha is h diffrnial quaion of h family of circls having hir cnrs

More information

Modelling of repairable items for production inventory with random deterioration

Modelling of repairable items for production inventory with random deterioration IOSR Journal of Mahmaics IOSR-JM -ISSN: 78-78, p-issn: 9-7X. Volum, Issu Vr. IV Jan - Fb., PP -9 www.iosrjournals.org Modlling of rpairabl ims for producion invnory wih random drioraion Dr.Ravish Kumar

More information

A MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA

A MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA MTHEMTICL MODEL FOR NTURL COOLING OF CUP OF TE 1 Mrs.D.Kalpana, 2 Mr.S.Dhvarajan 1 Snior Lcurr, Dparmn of Chmisry, PSB Polychnic Collg, Chnnai, India. 2 ssisan Profssor, Dparmn of Mahmaics, Dr.M.G.R Educaional

More information

Chapter 9 Review Questions

Chapter 9 Review Questions Chapr 9 Rviw Qusions. Using h - modl, show ha if marks clar and agns hav raional xpcaions hn mporary shocks canno hav prsisn ffcs on oupu. If marks clar and agns hav raional xpcaions hn mporary produciviy

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

ERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012

ERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012 ERROR AALYSIS AJ Pinar and D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr, 0 OVERVIEW Exprimnaion involvs h masurmn of raw daa in h laboraory or fild I is assumd

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

Chapter 6 Differential Equations and Mathematical Modeling

Chapter 6 Differential Equations and Mathematical Modeling 6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.

More information

MEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control

MEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion

More information

Discussion 06 Solutions

Discussion 06 Solutions STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P

More information

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

Ma/CS 6a Class 15: Flows and Bipartite Graphs

Ma/CS 6a Class 15: Flows and Bipartite Graphs //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d

More information