Macroeconomics 1. Ali Shourideh. Final Exam
|
|
- Tracey Dayna Washington
- 5 years ago
- Views:
Transcription
1 Macroeconomic 1 Ali Shourideh Final Exam Problem 1. A Model of On-he-Job Search Conider he following verion of he McCall earch model ha allow for on-he-job-earch. In paricular, uppoe ha ime i coninuou and worker - employed or unemployed dicoun he fuure a rae ρ. A worker ha ha a job eparae from her job a rae η. An unemployed worker find a job a rae λ 0 - he wage i hen drawn from a diribuion F w) which i coninuou. Moreover, a worker who ha a job ge an opporuniy o draw a new wage offer a rae λ 1 - again from he ame diribuion F w). The worker hen decide wheher o accep hi new offer or no. A uual, aume ha he flow value of unemploymen i b; he worker are rik-neural and canno ave. a. Le V w) be he value of a job a hand ha pay he wage w. Le U be he value of unemploymen. Wrie he Bellman equaion for U and V w). Soluion. The value funcion are given by ρv w) = w + λ 1 max {V w ) V w), 0} df w ) + η U V w)) 1) ρu = b + λ 0 max {V w) U, 0} df w) ) b. Calculae he reervaion wage, w, for an unemployed worker a a funcion of he value funcion V w). Under wha condiion, w < b hold? Provide an inuiion for hi reul. Soluion. For w, we mu have ha U = V w ). Subracing, ) from 1) evaluaed a w, we have w = b + λ 0 λ 1 ) max {V w) V w ), 0} df w) When λ 0 < λ 1, he above how ha w < b. Inuiively, when λ 1 > λ 0, acceping a job give a beer opion value o a worker han being unemployed - a higher probabiliy of a new draw which alway increae he value for a worker. Thi implie ha an unemployed worker i willing o ake a job a a wage lower han unemploymen benefi in order o obain hi beer opion value. c. Wha i he reervaion wage for an employed worker? Soluion. Since V w) i an increaing funcion of w, he reervaion wage of an employed worker i w - he wage of he job he i currenly employed in. d. How doe an increae in unemploymen benefi, b, affec he job-o-job urnover? Show your reul and explain inuiively. 1
2 Soluion. We have ρ + η) V w) = w + λ 1 ρ + η) V w) = 1 λ 1 V w) 1 F w)) V 1 w) = ρ + η + λ 1 1 F w)) w V w ) V w) df w ) + ηu Therefore, w = b + λ 0 λ 1 ) V w) V w ) df w) w = b λ 0 λ 1 ) V w) V w ) d 1 F w)) w = b + λ 0 λ 1 ) 1 F w)) V w) dw w = b + λ 0 λ 1 ) w 1 F w) ρ + η + λ 1 1 F w)) dw We have w b w b = 1 λ 0 λ 1 ) w b 1 + λ 0 λ 1 ) 1 F w )) = 1 ρ + η + λ 1 1 F w )) w ρ + η + λ0 1 F w )) = 1 b ρ + η + λ 1 1 F w )) 1 F w ) ρ + η + λ 1 1 F w )) Thi implie ha w > 0. A rie in he reervaion wage lead o a lower urnover a le people b accep job and herefore le people move beween job a well.
3 Problem. A Sochaic AK economy Conider an economy in which producion i done only wih capial and oupu i given by Y = A K where A i an i.i.d. ochaic proce and i diribued according o he c.d.f. F A). Capial depreciae a rae δ. Suppoe ha he economy exhibi a repreenaive agen and ha preference of uch hypoheical agen i given by β u C ) where u ) i a CRRA uiliy funcion given by { c 1 γ γ 1 1 γ u c) = log c γ = 1 =0 a. Define a Compeiive Equilibrium for hi economy auming ha rading occur in equenial ae marke. { Soluion. A compeiive equilibrium for hi economy i a equence of allocaion C ), X ), K +1 ), B a well a price {p ), r ), q )} where = A and: 1. Houehold olve: ubjec o p ) C ) + X ) + max β π ) u C )) =0 +1 q ) B ) r ) K 1 ) + p ) B ) 1 δ) K 1 ) + X ) = K +1 ). Firm olve 3. Marke clear max p ) A ) k r ) k k B ) = 0 ) = K ) 1 K f C ) + X ) = A ) K 1 ) Noe ha if A ha a coninuou diribuion, he above really hould be re-wrien wih inegral over inead of um. 3
4 b. Sae he Fir Welfare Theorem and ue i o formulae a compeiive equilibrium a a oluion o a planning problem. Soluion. According o he FWT, a compeiive equilibrium mu be pareo opimal. A a reul, i mu olve he following planning problem max β π ) u C )) ubjec o =0 C ) + K +1 ) = A ) K 1 ) + 1 δ) K 1 ) c. Formulae he planning problem problem recurively. You are required o ue only one ae variable. Soluion. Bellman equaion i given by V Y ) = max u C) + β ubjec o V A + 1 δ) K ) df A ) C + K = Y d. Solve he funcional equaion aociaed wih hi recurive problem and find he value funcion and policy funcion uing a gue and verify mehod. Wha aumpion on he fundamenal hould be made o ha he oluion o hi problem exi and i unique? Soluion. We gue ha V Y ) = B Y 1 γ. Wih hi gue, he opimizaion in he FE above 1 γ become Y K ) 1 γ A + 1 δ) K ) 1 γ max + βb df A ) K If we le x = K /Y, hen olving he above opimizaion i equivalen o 1 x) 1 γ A + 1 δ) x) 1 γ max + βb df A ) x Taking FOC, we have 1 x) γ = βbx γ E A + 1 δ) 1 γ ) γ x = βbe A + 1 δ) 1 γ 1 x For each B he above ha a unique oluion. We can herefore, replacing in he objecive in he FE and have v Y ) = C)1 γ A + βb + 1 δ) K ) 1 γ df A ) = Y 1 γ 1 x) 1 γ + βbx 1 γ E A + 1 δ) 1 γ = Y 1 γ 1 x) 1 γ + x 1 x) γ = Y 1 γ 1 x) γ = βbx γ E A + 1 δ) 1 γ Y 1 γ 4
5 which ha he form a he gueed value. If we e he above equal o B Y 1 γ, we have 1 γ βx γ E A + 1 δ) 1 γ = 1 x = βe A + 1 δ) 1 γ) 1 γ and B = 1 x) γ = 1 βe A + 1 δ) 1 γ) 1 ) γ γ Therefore, in order for hi oluion o be meaningful, we mu have ha βe A + 1 δ) 1 γ < 1. The policy funcion ha he form K = xy = x A + 1 δ) K e. Uing he policy funcion calculaed above, calculae he average growh rae of hi economy - auming ha he daa come from he aionary diribuion of he model. The reul hould be expreed in erm of he fundamenal parameer of he model a well a variou momen of he diribuion F ). Soluion. The growh rae of he economy i given by 1 + g +1 = A +1K +1 A K = A +1x A + 1 δ) K A K = x A +1 A + 1 δ) A Since hock are i.i.d., hi average growh rae i given by A + 1 δ 1 + g = xe A E A = βe A + 1 δ) 1 γ) 1 γ A + 1 δ E A E A In wha follow, aume ha δ = 1 and ha F A) = A 0 1 e log x+σ σx π / log A) Tha i, A i diribued ) according o a log-normal diribuion whoe mean i A and i variance i e σ 1 A. f. Wha i he average growh rae of he economy? How doe growh inerac wih rik, i.e., how doe he average growh rae of he economy depend on σ? Explain your anwer inuiively. σ dx 5
6 Soluion. From above, we have Noe ha log A N Therefore, 1 + g = βe A 1 γ) 1 γ E A log A σ, σ ). Therefore, E A 1 γ = E e 1 γ) log A 1 γ)γ 1 γ) log A+ σ = e 1 + g = β 1 1 γ A γ e 1 γ)σ Therefore, an increae in rik σ lead o an increae in growh when γ < 1 and a decreae in growh when γ > 1. The inuiion for hi reul can be een by rewriing he Euler equaion a follow: u C ) = βe A +1 u C +1 ) = β {E A +1 E u C +1 ) + Cov A +1, u C +1 ))} An increae in rik, ha wo effec: 1. i increae he average marginal uiliy E u C +1 ). Thi i becaue marginal uiliy i a convex funcion of conumpion and an increae in rik raie i average value - i increae he probabiliy of ail even. The conumer would like o inve more o inure hemelve again he rik of uch even a a precauion.. i decreae he correlaion beween marginal uiliy and reurn; ha i, i make i more negaive - uual rikaverion. When γ < 1, he precauionary moive dominae while when γ > 1, he rik-averion moive dominae. Thu for low value of γ, an increae in rik lead o higher invemen and higher growh while for high value of γ an increae in rik lead o low invemen and growh. g. Calculae he price of a rik-free real bond, P f. Wha i he rik-free rae in hi economy. Soluion. The price of a rik-free bond i given by The reurn on he ae i given by P f = βe u C +1 ) u C ) = βe 1 x) xa +1 A K ) γ 1 x) A K ) γ = βx γ E A γ +1 = β βe A 1 γ) γ γ E A γ = E A γ E A 1 γ R f = 1 P f = E A1 γ E A γ 6
7 h. Calculae he price, P, of holding a firm ha own he phyical capial and inve from he oupu ha i produce and how ha P = K ; Hin: Thi i an ae wih a dividend proce given by D = A K K +1. Wha i he reurn on uch an ae? Soluion. We have u C +1 ) P = βe u C ) D +1 + P +1 1 C γ = βe β D =+1 = βe =+1 +1 = βe A +1 K +1 + βe =+ +1 = βe A +1 K +1 + β 1 = βe 1 C γ β A K K +1 ) 1 C γ β A K 1 β 1 C γ K ) ) E β 1 β C γ A K K = A +1 K +1 + β 1 E β C 1K γ By he Euler equaion, we have E 1 β C γ 1 =+ A = 1 E 1 E =+ β C γ 1 β C γ 1 A 1 = 0 β 1K E 1 β C γ 1 A 1 = 0 β C γ 1 ) A 1 where he la equaliy follow from law of ieraed expecaion. We herefore have P = K +1 + β 1 ) E A 1 = K +1 which complee he claim. The reurn on he capial i given by R k +1 = P +1 + D +1 P = A +1K +1 K + + K + K +1 = A +1 7
8 i. Calculae he equiy premium a he difference beween average reurn on phyical capial and he rik-free rae. How doe hi depend on rik-averion, γ, and variance of he hock, σ. Explain inuiively. Soluion. The equiy premium i defined a E R k R = EA E A 1 γ E A γ = EAE A γ E A 1 γ E A γ = Ae γ log A σ )+ γ σ e 1 γ) e γ log A σ ) + γ σ = A1 γ e γ1+γ)σ A 1 γ e 1 γ)γσ A 1 γ e γ1+γ)σ = A 1 e γσ log A )+ σ 1 γ) σ Thi i an increaing funcion of γ and σ. A σ rie, he rik inheren in holding phyical capial increae and a a reul inveor will require a higher rae of reurn. The ame logic hold for γ; a houehold become more rik-avere hey demand a higher average rae of reurn in order o hold he capial ock. 8
Suggested Solutions to Midterm Exam Econ 511b (Part I), Spring 2004
Suggeed Soluion o Miderm Exam Econ 511b (Par I), Spring 2004 1. Conider a compeiive equilibrium neoclaical growh model populaed by idenical conumer whoe preference over conumpion ream are given by P β
More informationANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER
John Riley 6 December 200 NWER TO ODD NUMBERED EXERCIE IN CHPTER 7 ecion 7 Exercie 7-: m m uppoe ˆ, m=,, M (a For M = 2, i i eay o how ha I implie I From I, for any probabiliy vecor ( p, p 2, 2 2 ˆ ( p,
More informationProblem Set #3: AK models
Universiy of Warwick EC9A2 Advanced Macroeconomic Analysis Problem Se #3: AK models Jorge F. Chavez December 3, 2012 Problem 1 Consider a compeiive economy, in which he level of echnology, which is exernal
More informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
More information1. 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 informationLecture 26. Lucas and Stokey: Optimal Monetary and Fiscal Policy in an Economy without Capital (JME 1983) t t
Lecure 6. Luca and Sokey: Opimal Moneary and Fical Policy in an Economy wihou Capial (JME 983. A argued in Kydland and Preco (JPE 977, Opimal governmen policy i likely o be ime inconien. Fiher (JEDC 98
More informationInternational Business Cycle Models
Inernaional Buine Cycle Model Sewon Hur January 19, 2015 Overview In hi lecure, we will cover variou inernaional buine cycle model Endowmen model wih complee marke and nancial auarky Cole and Obfeld 1991
More informationMidterm Exam. Macroeconomic Theory (ECON 8105) Larry Jones. Fall September 27th, Question 1: (55 points)
Quesion 1: (55 poins) Macroeconomic Theory (ECON 8105) Larry Jones Fall 2016 Miderm Exam Sepember 27h, 2016 Consider an economy in which he represenaive consumer lives forever. There is a good in each
More informationEssential 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 informationCooperative Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS. August 8, :45 a.m. to 1:00 p.m.
Cooperaive Ph.D. Program in School of Economic Sciences and Finance QUALIFYING EXAMINATION IN MACROECONOMICS Augus 8, 213 8:45 a.m. o 1: p.m. THERE ARE FIVE QUESTIONS ANSWER ANY FOUR OUT OF FIVE PROBLEMS.
More information1 Answers to Final Exam, ECN 200E, Spring
1 Answers o Final Exam, ECN 200E, Spring 2004 1. A good answer would include he following elemens: The equiy premium puzzle demonsraed ha wih sandard (i.e ime separable and consan relaive risk aversion)
More informationCHAPTER 7: UNCERTAINTY
Eenial Microeconomic - ecion 7-3, 7-4 CHPTER 7: UNCERTINTY Fir and econd order ochaic dominance 2 Mean preerving pread 8 Condiional ochaic Dominance 0 Monoone Likelihood Raio Propery 2 Coninuou diribuion
More information1 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 informationFinancial 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 informationAdditional Methods for Solving DSGE Models
Addiional Mehod for Solving DSGE Model Karel Meren, Cornell Univeriy Reference King, R. G., Ploer, C. I. & Rebelo, S. T. (1988), Producion, growh and buine cycle: I. he baic neoclaical model, Journal of
More informationT. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 2011 EXAMINATION
ECON 841 T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 211 EXAMINATION This exam has wo pars. Each par has wo quesions. Please answer one of he wo quesions in each par for a
More informationProblem 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 informationLecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model
Lecure Noes 3: Quaniaive Analysis in DSGE Models: New Keynesian Model Zhiwei Xu, Email: xuzhiwei@sju.edu.cn The moneary policy plays lile role in he basic moneary model wihou price sickiness. We now urn
More informationThe 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 informationNotes on cointegration of real interest rates and real exchange rates. ρ (2)
Noe on coinegraion of real inere rae and real exchange rae Charle ngel, Univeriy of Wiconin Le me ar wih he obervaion ha while he lieraure (mo prominenly Meee and Rogoff (988) and dion and Paul (993))
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationInvestment-specific Technology Shocks, Neutral Technology Shocks and the Dunlop-Tarshis Observation: Theory and Evidence
Invemen-pecific Technology Shock, Neural Technology Shock and he Dunlop-Tarhi Obervaion: Theory and Evidence Moren O. Ravn, European Univeriy Iniue and he CEPR Saverio Simonelli, European Univeriy Iniue
More information( ) (, ) F K L = F, Y K N N N N. 8. Economic growth 8.1. Production function: Capital as production factor
8. Economic growh 8.. Producion funcion: Capial as producion facor Y = α N Y (, ) = F K N Diminishing marginal produciviy of capial and labor: (, ) F K L F K 2 ( K, L) K 2 (, ) F K L F L 2 ( K, L) L 2
More informationMacroeconomics Qualifying Examination
Macroeconomics Qualifying Examinaion January 205 Deparmen of Economics UNC Chapel Hill Insrucions: This examinaion consiss of four quesions. Answer all quesions. If you believe a quesion is ambiguously
More informationE β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.
Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke
More information1) According to the article, what is the main reason investors in US government bonds grow less optimistic?
14.02 Quiz 3 Soluion Fall 2004 Muliple-Choice Queion 1) According o he aricle, wha i he main reaon inveor in US governmen bond grow le opimiic? A) They are concerned abou he decline (depreciaion) of he
More informationThe 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 informationEconomics 6130 Cornell University Fall 2016 Macroeconomics, I - Part 2
Economics 6130 Cornell Universiy Fall 016 Macroeconomics, I - Par Problem Se # Soluions 1 Overlapping Generaions Consider he following OLG economy: -period lives. 1 commodiy per period, l = 1. Saionary
More informationANSWERS 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 informationIntroduction 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 informationProblem 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 informationIntermediate Macro In-Class Problems
Inermediae Macro In-Class Problems Exploring Romer Model June 14, 016 Today we will explore he mechanisms of he simply Romer model by exploring how economies described by his model would reac o exogenous
More informationFinal 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 informationLecture 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 informationARTIFICIAL INTELLIGENCE. Markov decision processes
INFOB2KI 2017-2018 Urech Univeriy The Neherland ARTIFICIAL INTELLIGENCE Markov deciion procee Lecurer: Silja Renooij Thee lide are par of he INFOB2KI Coure Noe available from www.c.uu.nl/doc/vakken/b2ki/chema.hml
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More informationEconomics 8105 Macroeconomic Theory Recitation 6
Economics 8105 Macroeconomic Theory Reciaion 6 Conor Ryan Ocober 11h, 2016 Ouline: Opimal Taxaion wih Governmen Invesmen 1 Governmen Expendiure in Producion In hese noes we will examine a model in which
More informationCOMPETITIVE 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 informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More informationLecture 19. RBC and Sunspot Equilibria
Lecure 9. RBC and Sunspo Equilibria In radiional RBC models, business cycles are propagaed by real echnological shocks. Thus he main sory comes from he supply side. In 994, a collecion of papers were published
More informationThe Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationUNIVERSITY 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 informationSolutions Problem Set 3 Macro II (14.452)
Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.
More informationThe 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 informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationResearch Article On Double Summability of Double Conjugate Fourier Series
Inernaional Journal of Mahemaic and Mahemaical Science Volume 22, Aricle ID 4592, 5 page doi:.55/22/4592 Reearch Aricle On Double Summabiliy of Double Conjugae Fourier Serie H. K. Nigam and Kuum Sharma
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationCONTROL SYSTEMS. Chapter 10 : State Space Response
CONTROL SYSTEMS Chaper : Sae Space Repone GATE Objecive & Numerical Type Soluion Queion 5 [GATE EE 99 IIT-Bombay : Mark] Conider a econd order yem whoe ae pace repreenaion i of he form A Bu. If () (),
More informationSimulating models with heterogeneous agents
Simulaing models wih heerogeneous agens Wouer J. Den Haan London School of Economics c by Wouer J. Den Haan Individual agen Subjec o employmen shocks (ε i, {0, 1}) Incomplee markes only way o save is hrough
More informationFINM 6900 Finance Theory
FINM 6900 Finance Theory Universiy of Queensland Lecure Noe 4 The Lucas Model 1. Inroducion In his lecure we consider a simple endowmen economy in which an unspecified number of raional invesors rade asses
More informationLars Nesheim. 17 January Last lecture solved the consumer choice problem.
Lecure 4 Locaional Equilibrium Coninued Lars Nesheim 17 January 28 1 Inroducory remarks Las lecure solved he consumer choice problem. Compued condiional demand funcions: C (I x; p; r (x)) and x; p; r (x))
More informationThe consumption-based determinants of the term structure of discount rates: Corrigendum. Christian Gollier 1 Toulouse School of Economics March 2012
The consumpion-based deerminans of he erm srucure of discoun raes: Corrigendum Chrisian Gollier Toulouse School of Economics March 0 In Gollier (007), I examine he effec of serially correlaed growh raes
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationReserves measures have an economic component eg. what could be extracted at current prices?
3.2 Non-renewable esources A. Are socks of non-renewable resources fixed? eserves measures have an economic componen eg. wha could be exraced a curren prices? - Locaion and quaniies of reserves of resources
More information4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
More informationThe Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationExplicit form of global solution to stochastic logistic differential equation and related topics
SAISICS, OPIMIZAION AND INFOMAION COMPUING Sa., Opim. Inf. Compu., Vol. 5, March 17, pp 58 64. Publihed online in Inernaional Academic Pre (www.iapre.org) Explici form of global oluion o ochaic logiic
More informationThis document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
More informationLABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012
LABOR MATCHING MODELS: BASIC DSGE IMPLEMENTATION APRIL 12, 2012 FIRM VACANCY-POSTING PROBLEM Dynamic firm profi-maimizaion problem ma 0 ( ) f Ξ v, n + 1 = 0 ( f y wn h g v ) Discoun facor beween ime 0
More informationSeminar 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 informationExplaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
More information13.3 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 informationFractional Ornstein-Uhlenbeck Bridge
WDS'1 Proceeding of Conribued Paper, Par I, 21 26, 21. ISBN 978-8-7378-139-2 MATFYZPRESS Fracional Ornein-Uhlenbeck Bridge J. Janák Charle Univeriy, Faculy of Mahemaic and Phyic, Prague, Czech Republic.
More informationAlgorithmic Trading: Optimal Control PIMS Summer School
Algorihmic Trading: Opimal Conrol PIMS Summer School Sebasian Jaimungal, U. Torono Álvaro Carea,U. Oxford many hanks o José Penalva,(U. Carlos III) Luhui Gan (U. Torono) Ryan Donnelly (Swiss Finance Insiue,
More informationWhat Ties Return Volatilities to Price Valuations and Fundamentals? On-Line Appendix
Wha Ties Reurn Volailiies o Price Valuaions and Fundamenals? On-Line Appendix Alexander David Haskayne School of Business, Universiy of Calgary Piero Veronesi Universiy of Chicago Booh School of Business,
More information1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
This documen was generaed a :45 PM 8/8/04 Copyrigh 04 Richard T. Woodward. An inroducion o dynamic opimizaion -- Opimal Conrol and Dynamic Programming AGEC 637-04 I. Overview of opimizaion Opimizaion is
More informationIntroduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.
Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since
More informationSuggested Solutions to Assignment 4 (REQUIRED) Submisson Deadline and Location: March 27 in Class
EC 450 Advanced Macroeconomics Insrucor: Sharif F Khan Deparmen of Economics Wilfrid Laurier Universiy Winer 2008 Suggesed Soluions o Assignmen 4 (REQUIRED) Submisson Deadline and Locaion: March 27 in
More informationExponential Sawtooth
ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able
More informationEXERCISES 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 information1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
This documen was generaed a :37 PM, 1/11/018 Copyrigh 018 Richard T. Woodward 1. An inroducion o dynamic opimiaion -- Opimal Conrol and Dynamic Programming AGEC 64-018 I. Overview of opimiaion Opimiaion
More informationProblem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.
CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex
More informationFull 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 informationSelfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos
Selfih Rouing Tim Roughgarden Cornell Univeriy Include join work wih Éva Tardo 1 Which roue would you chooe? Example: one uni of raffic (e.g., car) wan o go from o delay = 1 hour (no congeion effec) long
More informationThis document was generated at 7:34 PM, 07/27/09 Copyright 2009 Richard T. Woodward
his documen was generaed a 7:34 PM, 07/27/09 Copyrigh 2009 Richard. Woodward 15. Bang-bang and mos rapid approach problems AGEC 637 - Summer 2009 here are some problems for which he opimal pah does no
More informationFinal Exam. Tuesday, December hours
San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all
More informationNECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY
NECESSARY AND SUFFICIENT CONDITIONS FOR LATENT SEPARABILITY Ian Crawford THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemmap working paper CWP02/04 Neceary and Sufficien Condiion for Laen
More information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
More informationChapter 7: Inverse-Response Systems
Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem
More informationpe pt dt = e pt Probabilty of death given survival till t : pe pt = p Expected life at t : pe(s t)p ds = e (s t)p t =
BLANCHARD Probabiliy of Deah: π () = pe p ; Probabily of living ill : Ω () = pe p d = e p Probabily of deah given survival ill : pe p = p e p Expeced life a : (s ) pe (s )p ds = p 1 Populaion normalized
More informationPolicy regimes Theory
Advanced Moneary Theory and Policy EPOS 2012/13 Policy regimes Theory Giovanni Di Barolomeo giovanni.dibarolomeo@uniroma1.i The moneary policy regime The simple model: x = - s (i - p e ) + x e + e D p
More informationSolutions - Midterm Exam
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERITY OF NEW MEXICO ECE-34: ignals and ysems ummer 203 PROBLEM (5 PT) Given he following LTI sysem: oluions - Miderm Exam a) kech he impulse response
More informationLecture 2D: Rank-Size Rule
Econ 460 Urban Economics Lecure 2D: Rank-Size Rule Insrucor: Hiroki Waanabe Summer 2012 2012 Hiroki Waanabe 1 / 56 1 Rank-Size Rule 2 Eeckhou 3 Now We Know 2012 Hiroki Waanabe 2 / 56 1 Rank-Size Rule US
More information! ln 2xdx = (x ln 2x - x) 3 1 = (3 ln 6-3) - (ln 2-1)
7. e - d Le u = and dv = e - d. Then du = d and v = -e -. e - d = (-e - ) - (-e - )d = -e - + e - d = -e - - e - 9. e 2 d = e 2 2 2 d = 2 e 2 2d = 2 e u du Le u = 2, hen du = 2 d. = 2 eu = 2 e2.! ( - )e
More informationProblem 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 informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationIMPLICIT 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 informationESTIMATES FOR THE DERIVATIVE OF DIFFUSION SEMIGROUPS
Elec. Comm. in Probab. 3 (998) 65 74 ELECTRONIC COMMUNICATIONS in PROBABILITY ESTIMATES FOR THE DERIVATIVE OF DIFFUSION SEMIGROUPS L.A. RINCON Deparmen of Mahemaic Univeriy of Wale Swanea Singleon Par
More informationExercises, Part IV: THE LONG RUN
Exercie, Par IV: THE LOG RU 4. The olow Growh Model onider he olow rowh model wihou echnoloy prore and wih conan populaion. a) Define he eady ae condiion and repreen i raphically. b) how he effec of chane
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2015 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationUtility maximization in incomplete markets
Uiliy maximizaion in incomplee markes Marcel Ladkau 27.1.29 Conens 1 Inroducion and general seings 2 1.1 Marke model....................................... 2 1.2 Trading sraegy.....................................
More informationPricing the American Option Using Itô s Formula and Optimal Stopping Theory
U.U.D.M. Projec Repor 2014:3 Pricing he American Opion Uing Iô Formula and Opimal Sopping Theory Jona Bergröm Examenarbee i maemaik, 15 hp Handledare och examinaor: Erik Ekröm Januari 2014 Deparmen of
More informationBlock Diagram of a DCS in 411
Informaion source Forma A/D From oher sources Pulse modu. Muliplex Bandpass modu. X M h: channel impulse response m i g i s i Digial inpu Digial oupu iming and synchronizaion Digial baseband/ bandpass
More informationUnemployment and Mismatch in the UK
Unemploymen and Mismach in he UK Jennifer C. Smih Universiy of Warwick, UK CAGE (Cenre for Compeiive Advanage in he Global Economy) BoE/LSE Conference on Macroeconomics and Moneary Policy: Unemploymen,
More informationExamples of Dynamic Programming Problems
M.I.T. 5.450-Fall 00 Sloan School of Managemen Professor Leonid Kogan Examples of Dynamic Programming Problems Problem A given quaniy X of a single resource is o be allocaed opimally among N producion
More informationStochastic Modelling in Finance - Solutions to sheet 8
Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump
More informationNews-generated dependence and optimal portfolios for n stocks in a market of Barndor -Nielsen and Shephard type.
New-generaed dependence and opimal porfolio for n ock in a marke of Barndor -Nielen and Shephard ype. Carl Lindberg Deparmen of Mahemaical Saiic Chalmer Univeriy of Technology and Göeborg Univeriy Göeborg,
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationA Risk-Averse Insider and Asset Pricing in Continuous Time
Managemen Science and Financial Engineering Vol 9, No, May 3, pp-6 ISSN 87-43 EISSN 87-36 hp://dxdoiorg/7737/msfe39 3 KORMS A Rik-Avere Inider and Ae Pricing in oninuou Time Byung Hwa Lim Graduae School
More informationLecture 3: Solow Model II Handout
Economics 202a, Fall 1998 Lecure 3: Solow Model II Handou Basics: Y = F(K,A ) da d d d dk d = ga = n = sy K The model soluion, for he general producion funcion y =ƒ(k ): dk d = sƒ(k ) (n + g + )k y* =
More information