HOMEWORK #1 This homework assignment is due at 5PM on Friday, November 3 in Marnix Amand s mailbox.
|
|
- Moris Collins
- 5 years ago
- Views:
Transcription
1 Econ 50a (second half) Yale University Fall 2006 Prof. Tony Smith HOMEWORK # This homework assignment is due at 5PM on Friday, November 3 in Marnix Amand s mailbox.. Consider a growth model with capital accumulation equation k t+ = f(k t ) if t is even and k t+ = g(k t ) if t is odd. Assume that: Solution We have: (i) f(0) = g(0) = 0. (ii) f (0)g (0) >. (iii) lim k f (g(k))g (k) < and lim k g (f(k))f (k) <. (iv) f and g are strictly increasing and strictly concave. Show that, from any initial condition k 0 > 0, there is global convergence to a twocycle in which k t oscillates between two values. How are these values determined? k 2t = g(f(k 2t 2 )) k 2t+ = f(g(k 2t )) Let h be defined as h(k) = g(f(k)) k for all k. Should the sequence k 2t be converging, the limit necessarily is a solution of h(k) = 0 (cf. your favorite math course on sequences). Using properties (i) and (ii), we have that h(0) = 0 and h is increasing at 0. By differentiating twice, h is strictly concave. Using property (iii), h < 0 for k sufficiently large, and hence lim k h(k) = because, by concavity, h is under its tangents. It is thus proved that h(k) = 0 has one solution, and only one (using concavity again), for k > 0, called ˆk. Furthermore, since g f is increasing, we have, for k < ˆk, g(f(k)) < g(f(ˆk)) = ˆk, and, for k > ˆk, g(f(ˆk)) < ˆk. So the sequence k 2t is increasing and bounded for k 0 < ˆk and decreasing and bounded for k 0 > ˆk. In both cases, k 2t is converging, necessarily toward ˆk which is the only possible limit. Same story for k 2t+, which converges to the unique solution of f(g(k)) = k. This the global convergence to the two-cycle.
2 2. Consider a neoclassical growth model similar to the one that we have discussed in lecture, but in which the level of technology oscillates deterministically between two values A H and A L, where A H > A L. In particular, period-t output y t equals A H F (k t ) if t is even and equals A L F (k t ) if t is odd. The planner ( Robinson Crusoe ) seeks to maximize t=0 βt u(c t ), given k 0 > 0, subject to the resource constraint that c t +k t+ = y t + ( δ)k t and to the nonnegativity constraint k t+ 0 for all t. (a) Formulate the planner s problem recursively. (Hint: Consider two value functions, one for periods in which the level of technology is high and one for periods in which the level of technology is low. Find a pair of Bellman equations that these functions must satisfy.) (b) Let the felicity function u be logarithmic, let y t = A t k α t, and assume that capital depreciates fully in one period (i.e., set δ = ). Use a guess-and-verify method to find the two value functions in part (a). Describe fully the dynamic behavior of the capital stock. Solution (a) When A = A H, the planner s problem is given by the following value function: () V H (K) = max K {u (A H F (K) + ( δ) K K ) + βv L (K )} when A = A L the planner s problem is going to be given by: (2) V L (K) = max K {u (A L F (K) + ( δ) K K ) + βv H (K )} (b) If u (c) = log c, δ = and F (K t ) = K a t, then the above value functions become: V H (K) = max{log (A H K a K ) + βv L (K )} K V L (K) = max{log (A L K a K ) + βv H (K )} K We know guess that the value functions are of the form: V H = E + F log K V L = G + J log K In that case, the equation above for V H becomes: (3) E + F log K = max K {log (A H K a K ) + β (G + J log K )}
3 The first-order conditions are: A H K a K + βj = 0 K βj = K A H K a K K = A H βjk a βjk g H (K) = K = βja HK a + βj Plugging the decision rule derived above back in the original equation: E + F log K = log (A H K a JA ) HβK a + β (G + J log JA ) HβK a + βj + βj = log A H + a log K log ( + βj) + βg + βj log JA H β Therefore: βj log ( + βj) + βja log K (4) E = log A H ( + βj) log ( + βj) + βg + βj log JA H β and: (5) F = a + βaj Similarly the value function when A = A L, is given by: (6) G + J log K = max K {log (A L K a K ) + β (E + F log K )} The first-order condition gives the following decision rule: (7) g L (K) = K = βf A LK a + βf Plugging the decision rule back into the equation above, we get: G + J log K = log(a L K a βf A LK (E a + βf ) + β + F log βf A ) LK a + βf = log A L + a log K log ( + βf ) + βe + βf log βf A L + βf a log K βf log ( + βf )
4 Therefore: (8) G = log A L ( + βf ) log ( + βf ) + βe + βf log βf A L and (9) J = a + βaf combining the expression above for J with the one we found for F, gives us: J = a + βa (a + βaf ) J = a + βa2 β 2 a 2 = a ( + βa) ( + βa) ( βa) = a βa and thus: (0) a F = a + βa βa = a βa = J Therefore the equation for E now becomes: E = ( log A H + βa ) ( log + βa ) + βg + βa βa βa βa log A H = βa log A H + βg + βa βa log βa βa βa log βa Let M = () βa βa log log. Then E becomes: βa βa βa βa E = βa log A H + βg + M βa βa = And similarly plugging in for E, in the equation above we had for G we have: (2) G = βa log A L + βe + M Thus solving the above system of 2 equations and 2 unknowns, we get: E = ( β 2) ( ( βa) (log A H + β log A L ) + M + βm ) G = ( β 2) ( ( βa) (log A L + β log A H ) + M + βm )
5 Therefore the decision rule when A = A H is given by: (3) g H (K) = βaa H K a and when A = A L, is given by: (4) g L (K) = βaa L K a We will now show that there is a global convergence to a two-cycle in which K t oscillates between two values. These 2 values are: when A = A H and similarly: K H = βaa H K a H K H = (βaa H ) a (5) K L = (βaa L ) a when A = A L. We will show that the assumptions of Question 2 of Homework # hold indeed in this case and thus there is a global convergence to a two-cycle. We have: g H (0) = g L (0) = 0 Moreover we have that: g H (K) = βa2 A H K a and g L (K) = βa2 A L K a and thus g H (0) = and g L (0) = since a < 0. Therefore: (6) g H (0) g L (0) = > Furthermore: g H (g L (K)) = β a+ a a+ A H A a LK a2 g H (g L (K)) = β a+ a a+3 A H A a LK a2 and: g H (g L (K)) g L (K) = β a+ a a+3 A H A a LK a2 βa 2 A L K a = = β a+2 a a+5 A H A a+ +a 2 L Ka2 and since a 2 + a 2 < 0 (remember 0 < a < ) we have that: (7) lim K g H (g L (K)) g L (K) = 0
6 Similarly: (8) g L (g H (K)) g H (K) = β a+2 a a+5 A L A a+ +a 2 H Ka2 and: (9) lim = K g L (g H (K)) g H (K) = 0 Finally g H and g L are strictly increasing and strictly concave 3. Consider a neoclassical growth model in which the felicity function u has constant elasticity of intertemporal substitution σ : u(c) = c σ σ, where σ > 0 and u(c) = log(c) if σ =. In addition, assume that f(k) = Ak α +( δ)k, where A > 0, α (0, ), and δ [0, ]. What happens to the speed of convergence to the steady state as u becomes linear (i.e., as σ approaches 0)? As f becomes linear (i.e., as α approaches )? Try to provide economic intuition for your findings. (Note: As discussed in Section 4.2 in Chapter 4 of the lecture notes, the speed of convergence near the steady state is inversely related to the slope of the decision rule at the steady state.) Solution Using the notes (section 4.2), especially equation 4.2 on p. 47 (simply use the algebraic expressions for u and f): for u becoming linear, we have λ 0, so convergence is immediate. The intuition is that with linear utility, there is hardly any intertemporal smoothing, so the agent immediately jumps to the equilibrium for α becoming linear, we have λ, so convergence is very slow (in fact, there is no convergence). The intuition is that there are no decreasing returns to scale for saving, so the agent doesn t keep investing until he reaches equilibrium. 4. Consider the planning problem for a basic finite-horizon neoclassical growth model: max {c t, k t+ } T t=0 T β t log(c t ), t=0 given k 0 = 0 and subject to the resource constraint that c t + k t+ = Akt α + ( δ)k t and to the nonnegativity constraint k t+ 0 for all t T. Set β = 0.95, δ = 0.,
7 and α = 0.4. Choose A so that the steady-state value of capital in the corresponding infinite-horizon model is 00. Solve the model numerically (say, in Matlab) using the shooting method described in lecture on October 23: start by guessing a value for k, solve for k 2 from the Euler equation at time 0, then solve for k 3 from the Euler equation at time, and so on, until k T + is found (T being the time horizon). Then vary k and repeat until the appropriate value of k T + (what is it?) is found. Find the lowest value for T such that the highest value of capital between periods 0 and T exceeds 90. Solution We first derive the Euler equation for this problem, then we solve the question numerically. We can solve for Euler equation either from nonlinear programming or dynamic programming method. Here we use dynamic programming. Define f (k t ) = Akt α + ( δ) k t. The recursive formulation for this problem is v t (k t ) = max k t+ ln (f (k t ) k t+ ) + βv t+ (k t+ ) Note that here value function depends on the time subscript t. F.O.C. for t T is From Envelope theorem, we have c t + βv t+ (k t+ ) = 0 v t (k t ) = c t f (k t ) Iterate forward and plug back time subscripts, we get the Euler equation β c t+ f (k t+ ) = c t Plug in c t = f (k t ) k t+ and f (k t ) = Ak α t + ( δ) k t, we have k t+2 = f (k t+ ) β (f (k t ) k t+ ) f (k t+ ) k t+2 = ( ) Akt+ α + ( δ) k t+ β (Ak α t + ( δ) k t k t+ ) ( Aαkt+ α + ( δ) ) k t+2 = A ( + αβ) kt+ α + ( δ) ( + β) k t+ β (Akt α + ( δ) k t ) ( Aαkt+ α + ( δ) ) We can see that it is a second-order nonlinear difference equation, which has boundary condition k 0 = 0, k T + = 0. Before we go on to the numerical step, we solve for A. The infinite horizon steady state k solves f (k ) = β A = β ( δ) α (k ) α =
8 Now we start to solve for it numerically by using shooting method. For an error bound k T + < 0.0, when time horizon T exceeds 37, the highest value of capital between periods 0 and T exceeds 90. The approximate period- capital stock for T = 37 is k = If you use grid point search, to get such a precision you have to define the step size of grid as fine as 0 7.
HOMEWORK #3 This homework assignment is due at NOON on Friday, November 17 in Marnix Amand s mailbox.
Econ 50a second half) Yale University Fall 2006 Prof. Tony Smith HOMEWORK #3 This homework assignment is due at NOON on Friday, November 7 in Marnix Amand s mailbox.. This problem introduces wealth inequality
More information1. Using the model and notations covered in class, the expected returns are:
Econ 510a second half Yale University Fall 2006 Prof. Tony Smith HOMEWORK #5 This homework assignment is due at 5PM on Friday, December 8 in Marnix Amand s mailbox. Solution 1. a In the Mehra-Prescott
More informationSuggested Solutions to Homework #3 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homework #3 Econ 5b (Part I), Spring 2004. Consider an exchange economy with two (types of) consumers. Type-A consumers comprise fraction λ of the economy s population and type-b
More informationECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2
ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 2 The due date for this assignment is Tuesday, October 2. ( Total points = 50). (Two-sector growth model) Consider the
More informationADVANCED MACROECONOMIC TECHNIQUES NOTE 3a
316-406 ADVANCED MACROECONOMIC TECHNIQUES NOTE 3a Chris Edmond hcpedmond@unimelb.edu.aui Dynamic programming and the growth model Dynamic programming and closely related recursive methods provide an important
More informationDynamic Optimization Using Lagrange Multipliers
Dynamic Optimization Using Lagrange Multipliers Barbara Annicchiarico barbara.annicchiarico@uniroma2.it Università degli Studi di Roma "Tor Vergata" Presentation #2 Deterministic Infinite-Horizon Ramsey
More informationSlides II - Dynamic Programming
Slides II - Dynamic Programming Julio Garín University of Georgia Macroeconomic Theory II (Ph.D.) Spring 2017 Macroeconomic Theory II Slides II - Dynamic Programming Spring 2017 1 / 32 Outline 1. Lagrangian
More informationSolutions for Homework #4
Econ 50a (second half) Prof: Tony Smith TA: Theodore Papageorgiou Fall 2004 Yale University Dept. of Economics Solutions for Homework #4 Question (a) A Recursive Competitive Equilibrium for the economy
More informationProblem Set #4 Answer Key
Problem Set #4 Answer Key Economics 808: Macroeconomic Theory Fall 2004 The cake-eating problem a) Bellman s equation is: b) If this policy is followed: c) If this policy is followed: V (k) = max {log
More informationLecture 5: Some Informal Notes on Dynamic Programming
Lecture 5: Some Informal Notes on Dynamic Programming The purpose of these class notes is to give an informal introduction to dynamic programming by working out some cases by h. We want to solve subject
More informationA simple macro dynamic model with endogenous saving rate: the representative agent model
A simple macro dynamic model with endogenous saving rate: the representative agent model Virginia Sánchez-Marcos Macroeconomics, MIE-UNICAN Macroeconomics (MIE-UNICAN) A simple macro dynamic model with
More informationSuggested Solutions to Homework #6 Econ 511b (Part I), Spring 2004
Suggested Solutions to Homework #6 Econ 511b (Part I), Spring 2004 1. (a) Find the planner s optimal decision rule in the stochastic one-sector growth model without valued leisure by linearizing the Euler
More informationLecture 6: Discrete-Time Dynamic Optimization
Lecture 6: Discrete-Time Dynamic Optimization Yulei Luo Economics, HKU November 13, 2017 Luo, Y. (Economics, HKU) ECON0703: ME November 13, 2017 1 / 43 The Nature of Optimal Control In static optimization,
More informationLecture 5 Dynamics of the Growth Model. Noah Williams
Lecture 5 Dynamics of the Growth Model Noah Williams University of Wisconsin - Madison Economics 702/312 Spring 2016 An Example Now work out a parametric example, using standard functional forms. Cobb-Douglas
More informationDYNAMIC LECTURE 5: DISCRETE TIME INTERTEMPORAL OPTIMIZATION
DYNAMIC LECTURE 5: DISCRETE TIME INTERTEMPORAL OPTIMIZATION UNIVERSITY OF MARYLAND: ECON 600. Alternative Methods of Discrete Time Intertemporal Optimization We will start by solving a discrete time intertemporal
More informationTOBB-ETU - Econ 532 Practice Problems II (Solutions)
TOBB-ETU - Econ 532 Practice Problems II (Solutions) Q: Ramsey Model: Exponential Utility Assume that in nite-horizon households maximize a utility function of the exponential form 1R max U = e (n )t (1=)e
More informationChapter 3. Dynamic Programming
Chapter 3. Dynamic Programming This chapter introduces basic ideas and methods of dynamic programming. 1 It sets out the basic elements of a recursive optimization problem, describes the functional equation
More informationSuggested Solutions to Problem Set 2
Macroeconomic Theory, Fall 03 SEF, HKU Instructor: Dr. Yulei Luo October 03 Suggested Solutions to Problem Set. 0 points] Consider the following Ramsey-Cass-Koopmans model with fiscal policy. First, we
More informationECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko
ECON 582: Dynamic Programming (Chapter 6, Acemoglu) Instructor: Dmytro Hryshko Indirect Utility Recall: static consumer theory; J goods, p j is the price of good j (j = 1; : : : ; J), c j is consumption
More informationIntroduction to Recursive Methods
Chapter 1 Introduction to Recursive Methods These notes are targeted to advanced Master and Ph.D. students in economics. They can be of some use to researchers in macroeconomic theory. The material contained
More informationLecture 3: Hamilton-Jacobi-Bellman Equations. Distributional Macroeconomics. Benjamin Moll. Part II of ECON Harvard University, Spring
Lecture 3: Hamilton-Jacobi-Bellman Equations Distributional Macroeconomics Part II of ECON 2149 Benjamin Moll Harvard University, Spring 2018 1 Outline 1. Hamilton-Jacobi-Bellman equations in deterministic
More informationEndogenous Growth. Lecture 17 & 18. Topics in Macroeconomics. December 8 & 9, 2008
Review: Solow Model Review: Ramsey Model Endogenous Growth Lecture 17 & 18 Topics in Macroeconomics December 8 & 9, 2008 Lectures 17 & 18 1/29 Topics in Macroeconomics Outline Review: Solow Model Review:
More informationDynamic Optimization Problem. April 2, Graduate School of Economics, University of Tokyo. Math Camp Day 4. Daiki Kishishita.
Discrete Math Camp Optimization Problem Graduate School of Economics, University of Tokyo April 2, 2016 Goal of day 4 Discrete We discuss methods both in discrete and continuous : Discrete : condition
More informationADVANCED MACRO TECHNIQUES Midterm Solutions
36-406 ADVANCED MACRO TECHNIQUES Midterm Solutions Chris Edmond hcpedmond@unimelb.edu.aui This exam lasts 90 minutes and has three questions, each of equal marks. Within each question there are a number
More informationThe economy is populated by a unit mass of infinitely lived households with preferences given by. β t u(c Mt, c Ht ) t=0
Review Questions: Two Sector Models Econ720. Fall 207. Prof. Lutz Hendricks A Planning Problem The economy is populated by a unit mass of infinitely lived households with preferences given by β t uc Mt,
More informationEconomics 210B Due: September 16, Problem Set 10. s.t. k t+1 = R(k t c t ) for all t 0, and k 0 given, lim. and
Economics 210B Due: September 16, 2010 Problem 1: Constant returns to saving Consider the following problem. c0,k1,c1,k2,... β t Problem Set 10 1 α c1 α t s.t. k t+1 = R(k t c t ) for all t 0, and k 0
More informationTopic 5: The Difference Equation
Topic 5: The Difference Equation Yulei Luo Economics, HKU October 30, 2017 Luo, Y. (Economics, HKU) ME October 30, 2017 1 / 42 Discrete-time, Differences, and Difference Equations When time is taken to
More informationLecture 2 The Centralized Economy: Basic features
Lecture 2 The Centralized Economy: Basic features Leopold von Thadden University of Mainz and ECB (on leave) Advanced Macroeconomics, Winter Term 2013 1 / 41 I Motivation This Lecture introduces the basic
More informationLecture 4: The Bellman Operator Dynamic Programming
Lecture 4: The Bellman Operator Dynamic Programming Jeppe Druedahl Department of Economics 15th of February 2016 Slide 1/19 Infinite horizon, t We know V 0 (M t ) = whatever { } V 1 (M t ) = max u(m t,
More information1 Jan 28: Overview and Review of Equilibrium
1 Jan 28: Overview and Review of Equilibrium 1.1 Introduction What is an equilibrium (EQM)? Loosely speaking, an equilibrium is a mapping from environments (preference, technology, information, market
More informationLecture 2 The Centralized Economy
Lecture 2 The Centralized Economy Economics 5118 Macroeconomic Theory Kam Yu Winter 2013 Outline 1 Introduction 2 The Basic DGE Closed Economy 3 Golden Rule Solution 4 Optimal Solution The Euler Equation
More informationMonetary Economics: Solutions Problem Set 1
Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of
More informationLecture 3: Growth Model, Dynamic Optimization in Continuous Time (Hamiltonians)
Lecture 3: Growth Model, Dynamic Optimization in Continuous Time (Hamiltonians) ECO 503: Macroeconomic Theory I Benjamin Moll Princeton University Fall 2014 1/16 Plan of Lecture Growth model in continuous
More informationEC9A2 Advanced Macro Analysis - Class #1
EC9A2 Advanced Macro Analysis - Class #1 Jorge F. Chávez University of Warwick October 29, 2012 Outline 1. Some math 2. Shocking the Solow model 3. The Golden Rule 4. CES production function (more math)
More informationNotes for ECON 970 and ECON 973 Loris Rubini University of New Hampshire
Notes for ECON 970 and ECON 973 Loris Rubini University of New Hampshire 1 Introduction Economics studies resource allocation problems. In macroeconomics, we study economywide resource allocation problems.
More information1 The Basic RBC Model
IHS 2016, Macroeconomics III Michael Reiter Ch. 1: Notes on RBC Model 1 1 The Basic RBC Model 1.1 Description of Model Variables y z k L c I w r output level of technology (exogenous) capital at end of
More informationMacroeconomics Qualifying Examination
Macroeconomics Qualifying Examination August 2015 Department of Economics UNC Chapel Hill Instructions: This examination consists of 4 questions. Answer all questions. If you believe a question is ambiguously
More informationDynamic Problem Set 1 Solutions
Dynamic Problem Set 1 Solutions Jonathan Kreamer July 15, 2011 Question 1 Consider the following multi-period optimal storage problem: An economic agent imizes: c t} T β t u(c t ) (1) subject to the period-by-period
More informationRamsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path
Ramsey Cass Koopmans Model (1): Setup of the Model and Competitive Equilibrium Path Ryoji Ohdoi Dept. of Industrial Engineering and Economics, Tokyo Tech This lecture note is mainly based on Ch. 8 of Acemoglu
More informationGrowth Theory: Review
Growth Theory: Review Lecture 1.1, Exogenous Growth Topics in Growth, Part 2 June 11, 2007 Lecture 1.1, Exogenous Growth 1/76 Topics in Growth, Part 2 Growth Accounting: Objective and Technical Framework
More informationLecture 1: Overview, Hamiltonians and Phase Diagrams. ECO 521: Advanced Macroeconomics I. Benjamin Moll. Princeton University, Fall
Lecture 1: Overview, Hamiltonians and Phase Diagrams ECO 521: Advanced Macroeconomics I Benjamin Moll Princeton University, Fall 2016 1 Course Overview Two Parts: (1) Substance: income and wealth distribution
More informationPublic Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model. Burkhard Heer University of Augsburg, Germany
Public Economics The Macroeconomic Perspective Chapter 2: The Ramsey Model Burkhard Heer University of Augsburg, Germany October 3, 2018 Contents I 1 Central Planner 2 3 B. Heer c Public Economics: Chapter
More informationAdvanced Macroeconomics
Advanced Macroeconomics The Ramsey Model Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 30 Introduction Authors: Frank Ramsey (1928), David Cass (1965) and Tjalling
More informationProjection Methods. Michal Kejak CERGE CERGE-EI ( ) 1 / 29
Projection Methods Michal Kejak CERGE CERGE-EI ( ) 1 / 29 Introduction numerical methods for dynamic economies nite-di erence methods initial value problems (Euler method) two-point boundary value problems
More informationUniversity of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming
University of Warwick, EC9A0 Maths for Economists 1 of 63 University of Warwick, EC9A0 Maths for Economists Lecture Notes 10: Dynamic Programming Peter J. Hammond Autumn 2013, revised 2014 University of
More informationUNIVERSITY OF VIENNA
WORKING PAPERS Cycles and chaos in the one-sector growth model with elastic labor supply Gerhard Sorger May 2015 Working Paper No: 1505 DEPARTMENT OF ECONOMICS UNIVERSITY OF VIENNA All our working papers
More informationNeoclassical Growth Model / Cake Eating Problem
Dynamic Optimization Institute for Advanced Studies Vienna, Austria by Gabriel S. Lee February 1-4, 2008 An Overview and Introduction to Dynamic Programming using the Neoclassical Growth Model and Cake
More informationECON 2010c Solution to Problem Set 1
ECON 200c Solution to Problem Set By the Teaching Fellows for ECON 200c Fall 204 Growth Model (a) Defining the constant κ as: κ = ln( αβ) + αβ αβ ln(αβ), the problem asks us to show that the following
More informationDynamic optimization: a recursive approach. 1 A recursive (dynamic programming) approach to solving multi-period optimization problems:
E 600 F 206 H # Dynamic optimization: a recursive approach A recursive (dynamic programming) approach to solving multi-period optimization problems: An example A T + period lived agent s value of life
More informationLecture 9. Dynamic Programming. Randall Romero Aguilar, PhD I Semestre 2017 Last updated: June 2, 2017
Lecture 9 Dynamic Programming Randall Romero Aguilar, PhD I Semestre 2017 Last updated: June 2, 2017 Universidad de Costa Rica EC3201 - Teoría Macroeconómica 2 Table of contents 1. Introduction 2. Basics
More informationLecture 10. Dynamic Programming. Randall Romero Aguilar, PhD II Semestre 2017 Last updated: October 16, 2017
Lecture 10 Dynamic Programming Randall Romero Aguilar, PhD II Semestre 2017 Last updated: October 16, 2017 Universidad de Costa Rica EC3201 - Teoría Macroeconómica 2 Table of contents 1. Introduction 2.
More informationChristiano 416, Fall 2014 Homework 1, due Wednesday, October Consider an economy in which household preferences have the following form:
Christiano 416, Fall 2014 Homework 1, due Wednesday, October 15. 1. Consider an economy in which household preferences have the following form: β t u(c t, n t ), β =.99, and u(c t, n t ) = log(c t ) +
More informationIntroduction to Real Business Cycles: The Solow Model and Dynamic Optimization
Introduction to Real Business Cycles: The Solow Model and Dynamic Optimization Vivaldo Mendes a ISCTE IUL Department of Economics 24 September 2017 (Vivaldo M. Mendes ) Macroeconomics (M8674) 24 September
More informationThe representative agent model
Chapter 3 The representative agent model 3.1 Optimal growth In this course we re looking at three types of model: 1. Descriptive growth model (Solow model): mechanical, shows the implications of a given
More informationModern Macroeconomics II
Modern Macroeconomics II Katsuya Takii OSIPP Katsuya Takii (Institute) Modern Macroeconomics II 1 / 461 Introduction Purpose: This lecture is aimed at providing students with standard methods in modern
More informationA Quick Introduction to Numerical Methods
Chapter 5 A Quick Introduction to Numerical Methods One of the main advantages of the recursive approach is that we can use the computer to solve numerically interesting models. There is a wide variety
More informationECON607 Fall 2010 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment 1 Suggested Solutions
ECON607 Fall 200 University of Hawaii Professor Hui He TA: Xiaodong Sun Assignment Suggested Solutions The due date for this assignment is Thursday, Sep. 23.. Consider an stochastic optimal growth model
More informationDynamic Programming Theorems
Dynamic Programming Theorems Prof. Lutz Hendricks Econ720 September 11, 2017 1 / 39 Dynamic Programming Theorems Useful theorems to characterize the solution to a DP problem. There is no reason to remember
More informationDynamic (Stochastic) General Equilibrium and Growth
Dynamic (Stochastic) General Equilibrium and Growth Martin Ellison Nuffi eld College Michaelmas Term 2018 Martin Ellison (Nuffi eld) D(S)GE and Growth Michaelmas Term 2018 1 / 43 Macroeconomics is Dynamic
More informationPermanent Income Hypothesis Intro to the Ramsey Model
Consumption and Savings Permanent Income Hypothesis Intro to the Ramsey Model Lecture 10 Topics in Macroeconomics November 6, 2007 Lecture 10 1/18 Topics in Macroeconomics Consumption and Savings Outline
More informationDynamic Macroeconomics: Problem Set 4
Dynamic Macroeconomics: Problem Set 4 Universität Siegen Dynamic Macroeconomics 1 / 28 1 Computing growth rates 2 Golden rule saving rate 3 Simulation of the Solow Model 4 Growth accounting Dynamic Macroeconomics
More informationLecture 1: Dynamic Programming
Lecture 1: Dynamic Programming Fatih Guvenen November 2, 2016 Fatih Guvenen Lecture 1: Dynamic Programming November 2, 2016 1 / 32 Goal Solve V (k, z) =max c,k 0 u(c)+ E(V (k 0, z 0 ) z) c + k 0 =(1 +
More informationTopic 2. Consumption/Saving and Productivity shocks
14.452. Topic 2. Consumption/Saving and Productivity shocks Olivier Blanchard April 2006 Nr. 1 1. What starting point? Want to start with a model with at least two ingredients: Shocks, so uncertainty.
More informationPractical Dynamic Programming: An Introduction. Associated programs dpexample.m: deterministic dpexample2.m: stochastic
Practical Dynamic Programming: An Introduction Associated programs dpexample.m: deterministic dpexample2.m: stochastic Outline 1. Specific problem: stochastic model of accumulation from a DP perspective
More informationDiscrete State Space Methods for Dynamic Economies
Discrete State Space Methods for Dynamic Economies A Brief Introduction Craig Burnside Duke University September 2006 Craig Burnside (Duke University) Discrete State Space Methods September 2006 1 / 42
More informationAdvanced Macroeconomics
Advanced Macroeconomics The Ramsey Model Micha l Brzoza-Brzezina/Marcin Kolasa Warsaw School of Economics Micha l Brzoza-Brzezina/Marcin Kolasa (WSE) Ad. Macro - Ramsey model 1 / 47 Introduction Authors:
More informationDevelopment Economics (PhD) Intertemporal Utility Maximiza
Development Economics (PhD) Intertemporal Utility Maximization Department of Economics University of Gothenburg October 7, 2015 1/14 Two Period Utility Maximization Lagrange Multiplier Method Consider
More informationThe Real Business Cycle Model
The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.
More informationNeoclassical Business Cycle Model
Neoclassical Business Cycle Model Prof. Eric Sims University of Notre Dame Fall 2015 1 / 36 Production Economy Last time: studied equilibrium in an endowment economy Now: study equilibrium in an economy
More informationMacro I - Practice Problems - Growth Models
Macro I - Practice Problems - Growth Models. Consider the infinitely-lived agent version of the growth model with valued leisure. Suppose that the government uses proportional taxes (τ c, τ n, τ k ) on
More informationLecture 4: Dynamic Programming
Lecture 4: Dynamic Programming Fatih Guvenen January 10, 2016 Fatih Guvenen Lecture 4: Dynamic Programming January 10, 2016 1 / 30 Goal Solve V (k, z) =max c,k 0 u(c)+ E(V (k 0, z 0 ) z) c + k 0 =(1 +
More informationPractice Questions for Mid-Term I. Question 1: Consider the Cobb-Douglas production function in intensive form:
Practice Questions for Mid-Term I Question 1: Consider the Cobb-Douglas production function in intensive form: y f(k) = k α ; α (0, 1) (1) where y and k are output per worker and capital per worker respectively.
More informationProblem Set #2: Overlapping Generations Models Suggested Solutions - Q2 revised
University of Warwick EC9A Advanced Macroeconomic Analysis Problem Set #: Overlapping Generations Models Suggested Solutions - Q revised Jorge F. Chavez December 6, 0 Question Consider the following production
More informationReal Business Cycle Model (RBC)
Real Business Cycle Model (RBC) Seyed Ali Madanizadeh November 2013 RBC Model Lucas 1980: One of the functions of theoretical economics is to provide fully articulated, artificial economic systems that
More informationLecture notes on modern growth theory
Lecture notes on modern growth theory Part 2 Mario Tirelli Very preliminary material Not to be circulated without the permission of the author October 25, 2017 Contents 1. Introduction 1 2. Optimal economic
More informationLecture 7: Linear-Quadratic Dynamic Programming Real Business Cycle Models
Lecture 7: Linear-Quadratic Dynamic Programming Real Business Cycle Models Shinichi Nishiyama Graduate School of Economics Kyoto University January 10, 2019 Abstract In this lecture, we solve and simulate
More informationMacroeconomics Qualifying Examination
Macroeconomics Qualifying Examination January 2016 Department of Economics UNC Chapel Hill Instructions: This examination consists of 3 questions. Answer all questions. If you believe a question is ambiguously
More informationMacroeconomic Topics Homework 1
March 25, 2004 Kjetil Storesletten. Macroeconomic Topics Homework 1 Due: April 23 1 Theory 1.1 Aggregation Consider an economy consisting of a continuum of agents of measure 1 who solve max P t=0 βt c
More informationUncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6
1 Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 A Two-Period Example Suppose the economy lasts only two periods, t =0, 1. The uncertainty arises in the income (wage) of period 1. Not that
More informationEquilibrium in a Production Economy
Equilibrium in a Production Economy Prof. Eric Sims University of Notre Dame Fall 2012 Sims (ND) Equilibrium in a Production Economy Fall 2012 1 / 23 Production Economy Last time: studied equilibrium in
More informationLecture 3 - Solow Model
Lecture 3 - Solow Model EC308 Advanced Macroeconomics 16/02/2016 (EC308) Lecture 3 - Solow Model 16/02/2016 1 / 26 Introduction Solow Model Sometimes known as Solow-Swan Model: Solow (1956): General Production
More informationEconomics 202A Lecture Outline #3 (version 1.0)
Economics 202A Lecture Outline #3 (version.0) Maurice Obstfeld Steady State of the Ramsey-Cass-Koopmans Model In the last few lectures we have seen how to set up the Ramsey-Cass- Koopmans Model in discrete
More informationAssumption 5. The technology is represented by a production function, F : R 3 + R +, F (K t, N t, A t )
6. Economic growth Let us recall the main facts on growth examined in the first chapter and add some additional ones. (1) Real output (per-worker) roughly grows at a constant rate (i.e. labor productivity
More informationLinear and Loglinear Approximations (Started: July 7, 2005; Revised: February 6, 2009)
Dave Backus / NYU Linear and Loglinear Approximations (Started: July 7, 2005; Revised: February 6, 2009) Campbell s Inspecting the mechanism (JME, 1994) added some useful intuition to the stochastic growth
More informationMacroeconomics Theory II
Macroeconomics Theory II Francesco Franco FEUNL February 2016 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 1 / 18 Road Map Research question: we want to understand businesses cycles.
More informationProblem Set 2: Proposed solutions Econ Fall Cesar E. Tamayo Department of Economics, Rutgers University
Problem Set 2: Proposed solutions Econ 504 - Fall 202 Cesar E. Tamayo ctamayo@econ.rutgers.edu Department of Economics, Rutgers University Simple optimal growth (Problems &2) Suppose that we modify slightly
More informationLecture notes for Macroeconomics I, 2004
Lecture notes for Macroeconomics I, 2004 Per Krusell Please do NOT distribute without permission! Comments and suggestions are welcome. 1 Chapter 3 Dynamic optimization There are two common approaches
More informationProblem 1 (30 points)
Problem (30 points) Prof. Robert King Consider an economy in which there is one period and there are many, identical households. Each household derives utility from consumption (c), leisure (l) and a public
More informationECON 582: The Neoclassical Growth Model (Chapter 8, Acemoglu)
ECON 582: The Neoclassical Growth Model (Chapter 8, Acemoglu) Instructor: Dmytro Hryshko 1 / 21 Consider the neoclassical economy without population growth and technological progress. The optimal growth
More informationproblem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
More informationHeterogeneous Agent Models: I
Heterogeneous Agent Models: I Mark Huggett 2 2 Georgetown September, 2017 Introduction Early heterogeneous-agent models integrated the income-fluctuation problem into general equilibrium models. A key
More informationPerturbation Methods I: Basic Results
Perturbation Methods I: Basic Results (Lectures on Solution Methods for Economists V) Jesús Fernández-Villaverde 1 and Pablo Guerrón 2 March 19, 2018 1 University of Pennsylvania 2 Boston College Introduction
More informationPart A: Answer question A1 (required), plus either question A2 or A3.
Ph.D. Core Exam -- Macroeconomics 5 January 2015 -- 8:00 am to 3:00 pm Part A: Answer question A1 (required), plus either question A2 or A3. A1 (required): Ending Quantitative Easing Now that the U.S.
More informationNew Notes on the Solow Growth Model
New Notes on the Solow Growth Model Roberto Chang September 2009 1 The Model The firstingredientofadynamicmodelisthedescriptionofthetimehorizon. In the original Solow model, time is continuous and the
More informationThe Ramsey Model. Alessandra Pelloni. October TEI Lecture. Alessandra Pelloni (TEI Lecture) Economic Growth October / 61
The Ramsey Model Alessandra Pelloni TEI Lecture October 2015 Alessandra Pelloni (TEI Lecture) Economic Growth October 2015 1 / 61 Introduction Introduction Introduction Ramsey-Cass-Koopmans model: di ers
More information0 β t u(c t ), 0 <β<1,
Part 2 1. Certainty-Equivalence Solution Methods Consider the model we dealt with previously, but now the production function is y t = f(k t,z t ), where z t is a stochastic exogenous variable. For example,
More informationLecture 15. Dynamic Stochastic General Equilibrium Model. Randall Romero Aguilar, PhD I Semestre 2017 Last updated: July 3, 2017
Lecture 15 Dynamic Stochastic General Equilibrium Model Randall Romero Aguilar, PhD I Semestre 2017 Last updated: July 3, 2017 Universidad de Costa Rica EC3201 - Teoría Macroeconómica 2 Table of contents
More informationMyopic and perfect foresight in the OLG model
Economics Letters 67 (2000) 53 60 www.elsevier.com/ locate/ econbase a Myopic and perfect foresight in the OLG model a b, * Philippe Michel, David de la Croix IUF, Universite de la Mediterranee and GREQAM,
More informationProjection Methods. Felix Kubler 1. October 10, DBF, University of Zurich and Swiss Finance Institute
Projection Methods Felix Kubler 1 1 DBF, University of Zurich and Swiss Finance Institute October 10, 2017 Felix Kubler Comp.Econ. Gerzensee, Ch5 October 10, 2017 1 / 55 Motivation In many dynamic economic
More informationLecture notes for Macroeconomics I, 2004
Lecture notes for Macroeconomics I, 2004 Per Krusell Please do NOT distribute without permission Comments and suggestions are welcome! 1 2 Chapter 1 Introduction These lecture notes cover a one-semester
More information