Chapter 9 Solow. O. Afonso, P. B. Vasconcelos. Computational Economics: a concise introduction
|
|
- Clifton Horn
- 5 years ago
- Views:
Transcription
1 Chapter 9 Solow O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 27
2 Overview 1 Introduction 2 Economic model 3 Computational implementation 4 Numerical results and simulation 5 Highlights 6 Main references O. Afonso, P. B. Vasconcelos Computational Economics 2 / 27
3 Introduction In 1956, Robert Solow published a seminal paper on economic growth and development. The proposed model, also known has the Solow Swan, ignores some important aspects of macroeconomics, such as short-run fluctuations in employment, and makes several assumptions to describe the long-run path of the economy. It remains highly influential even today and despite its relative simplicity conveys a number of very useful insights about the dynamics of the growth process. It is also worth teaching from a methodological perspective. Numerical methods for solving differential equations will be introduced and the Euler method implemented to solve initial value problems (Dahlquist and Björck (2008) and Süli and Mayers (2003)). O. Afonso, P. B. Vasconcelos Computational Economics 3 / 27
4 Economic model This is a model of capital accumulation in a pure production economy. Countries produce and consume only a single, homogeneous good (output, GDP or real income). There are no prices because there is no need for money. Everyone works all the time, saves a fixed portion of income, invests and owns the firm (consumer side is not modelled). There is no government, thus no taxation nor subsidies. It is a closed economy model. This model captures the pure impact that savings (investment) has on the long-run standard of living (per capita income). The model is built around two equations: a production function and a capital accumulation equation. O. Afonso, P. B. Vasconcelos Computational Economics 4 / 27
5 Economic model Production function The neoclassic production function is assumed to have the Cobb Douglas form: Y (t) = F (K (t), L(t)) = K (t) α L(t) 1 α (1) where K (t) is capital input, L(t) is labour input and α, between 0 and 1, is the capital share in production. This production function is neoclassic since it exhibits constant returns to scale, presents positive and diminishing marginal returns to factor accumulation, satisfies the Inada conditions. O. Afonso, P. B. Vasconcelos Computational Economics 5 / 27
6 Economic model Production function In per capita terms, the production function (1) is: where y = Y (t) L(t) Y (t) L(t) = K (t)α L(t) 1 α y(t) = k(t) α (2) L(t) and k(t) = K (t) L(t) From (1), taking logs and differentiating with respect to time on both sides, denoting by Ẏ (t) and L(t) the derivatives of Y and L with respect to t, the results is Ẏ (t) Y (t) = α K (t) L(t) + (1 α) K (t) L(t). The growth rate of per capita output is then simply ẏ(t) y(t) = α k(t) k(t). Thus, the source of increase in per capita output is the capital deepening. O. Afonso, P. B. Vasconcelos Computational Economics 6 / 27
7 Capital accumulation Economic model The second key equation describes the path of capital accumulation K (t) = Y (t) C(t) δk (t) (3) i.e. the addition to the capital stock each period, K (t), depends positively on savings and negatively on depreciation, which takes place at rate δ. Since a fraction s of output is saved, Y (t) C(t) = sy (t), the labour input grows at rate L(t) L(t) = n, the path of per capita capital accumulation is k(t) = sy(t) (δ + n)k(t). (4) Thus, the change in per capita capital each period is determined by three terms: per capita investment, sy(t); per capita depreciation, δk(t); and population growth, nk(t). This is a differential equation and, for an initial stock of capital k 0, it defines an initial value problem. O. Afonso, P. B. Vasconcelos Computational Economics 7 / 27
8 Economic model Transitional dynamics: the Solow diagram The Solow diagram has two curves, plotted as functions of k(t), and can be used to understand per capita output evolves over time f(k) consumption (δ+n)k sf(k) investment k Solow diagram O. Afonso, P. B. Vasconcelos Computational Economics 8 / 27
9 Economic model Transitional dynamics: the Solow diagram The curve y(t) = f (k(t)) = k(t) α is the production function, and the curve sy(t) = sk(t) α depicts the amount of per capita investment, which shifts y(t) down by the factor s. In turn, the line (δ + n)k(t) represents the amount of new per capita investment required to keep the amount of per capita capital constant. When the difference between the curve sy(t) and the line (δ + n)k(t) is positive (negative) the economy is increasing (decreasing) its per capita capital: capital deepening (widening) is occurring. when sy(t) = (n + δ)k(t), so that k(t) = 0, the amount of per capita capital remains constant, and such a point is a steady state. To sum up, we have the following. From the diagram it is found the steady-state value of per capita capital. Then, from the production function, the also constant steady-state value of output per worker is obtained. In turn, the steady state per capita consumption is given by the difference between steady state per capita values of output and investment. O. Afonso, P. B. Vasconcelos Computational Economics 9 / 27
10 Economic model Steady state Since, in steady state, k(t) = 0, (4) and (2) can be used to solve the steady-state values of per capita capital and per capita output; thus, k(t) = sk(t) α (δ + n)k(t) 0 = sk α (δ + n)k and so the steady-state values are: k = ( s ) 1 1 α δ + n (5) y = ( ) α s 1 α. (6) δ + n Ceteris paribus, countries that have high savings/investment rates will tend to be richer since they accumulate more per capita capital and, as a result, have more per capita output. In turn, countries that have high population growth rates will tend to be poorer. O. Afonso, P. B. Vasconcelos Computational Economics 10 / 27
11 Economic model Golden rule The optimal capital accumulation leads to the golden rule savings rate that maximises the steady-state level of consumption. The per capita consumption c(t) is given by c(t) = (1 s)y(t) and s can be written as s = y(t) c(t) y(t). At the steady state, k(t) = 0, that is, sy = (δ + n)k. The latter can be written in terms of c(t), as c = y (δ + n)k. To find k that maximises c, the first order condition, dc dk = 0, produces k gr = considering A = 1. Note that k > k gr whenever s > α. ( ) 1 α 1 α, (7) δ + n O. Afonso, P. B. Vasconcelos Computational Economics 11 / 27
12 Economic model Golden rule f(k) (δ+n)k sf(k) s gr f(k) k gr k Golden rule savings rate O. Afonso, P. B. Vasconcelos Computational Economics 12 / 27
13 Linear approximation Economic model The Solow model has a closed solution and can be demonstrated that the convergence process is (globally) stable. Taking (4), considering k(t) = φ(k(t)), a first order Taylor approximation for k(t), being φ(k(t)) at least twice differentiable, is given by k(t) φ(k ) + dφ(k(t)) dk(t) k (k(t) k ) ( ) sf (k ) (n + δ) (k(t) k ) since φ(k ) = 0. Thus, whenever k(t) < k (k(t) > k ) then k(t) > 0 ( k(t) < 0) and physical capital will accumulate (diminish) towards the steady state. O. Afonso, P. B. Vasconcelos Computational Economics 13 / 27
14 Economic model Differential equation Since most differential equations are not analytically soluble, numerical solution of ordinary differential equations is a fundamental technique. A differential equation involves an unknown function, y(t), and its derivatives. A first order ordinary differential equation, ODE, has the form where f : R R R and y(t) : [t 0, t T ] R R. ẏ(t) = dy (t) = f (t, y(t)) (8) dt Equation (8) is non-autonomous but often economic problems are time-autonomous ẏ(t) = dy (t) = f (y(t)). (9) dt The solution of (8) (or (9)) is a family of functions determined by a constant. O. Afonso, P. B. Vasconcelos Computational Economics 14 / 27
15 Initial value problems Economic model A particular solution is computed by requiring that it goes through a specific point, the initial condition, (t 0, y 0 = y(t 0 )). The problem specified both by (8) (or (9)) and the initial condition is called an initial value problem, IVP. Solving the IVP is to predict the path that a quantity will take during a certain time interval, given the initial quantity. Problems involving ODEs of a higher order can be reduced to a system of first ODE equations by introducing new variables. Seldom do these equations have solutions that can be expressed in a closed form or the analytical form is often too cumbersome; solution techniques are generally unable to deal with large and nonlinear systems of equations that arise in real problems. O. Afonso, P. B. Vasconcelos Computational Economics 15 / 27
16 Initial value problems Economic model To solve a continuous problem in a computer, a discretisation process is required. Initial value problems can be numerically solved using finite difference methods and recursive procedures. The numerical procedures are based on approximations y 0, y 1,, y T to the exact solution y(t 0 ), y(t 1 ),, y(t T ) at the grid points: t 0 < t 1 < < t T. The distances h n = t n t n 1, n = 1,, T, are called step sizes and, for simplicity, equal step sizes, that is, uniform grids, where h = (t T t 0 )/T, are considered. The aim is, starting with the initial value y 0 = y(t 0 ), to find y n, which approximates y(t n ), by recurrence relations in such a way that the value of y n+1 could be stated as a function of y n. O. Afonso, P. B. Vasconcelos Computational Economics 16 / 27
17 Economic model Euler method From (8), the slope of the tangent line at (t 0, y(t 0 )) can be computed; taking a small step along the computed tangent, another point (t 1, y 1 ) is reached, from where again a tangent can be computed; perform this process iteratively until (t T, y T ); a polygonal curve y 0, y T is computed, approximating the sought solution: y(t 0 ), y(t T ) at points t 0, t T. The approximation error can be reduced considering small step sizes, h. This is know as the Euler method y n+1 = y n + hf (t n, y n ), n = 0, 1,, T. (10) It is a first-order method since the error of the final result, global error, is proportional to h. The error in a single step, local error, is proportional to h 2. High-order numerical methods will be explored in the following chapters. O. Afonso, P. B. Vasconcelos Computational Economics 17 / 27
18 Economic model It is relevant for the stability of (linearised) economic models to get some insight on the solution of linear differential equations. The solution of ẏ(t) = ay(t) + b can be obtained from the solution of the homogeneous equations (b = 0) and the particular solution. The stability behaviour is ruled by the former. Since e t satisfies ẏ(t) = y(t), one should try e λt. Then, we get the result that λ = a and thus y(t) = ce at, c constant (which can be determined for an initial value problem). The stability of the solution depends on the sign of λ: if λ < 0 then the solution converges to the steady state. It is worth mentioning that the behaviour is different from the discrete case (e λt vs λ t ): for discrete time both sign and modulus of λ are relevant to understand the equations behaviour. O. Afonso, P. B. Vasconcelos Computational Economics 18 / 27
19 Computational implementation The following baseline values are considered: s = 0.4, A = 1, α = 0.3, δ = 0.1 and n = O. Afonso, P. B. Vasconcelos Computational Economics 19 / 27
20 Computational implementation Presentation and parameters %% Solow model % Neoclassical growth model ( exogenous growth model ) % Implemented by : P. B. Vasconcelos and O. Afonso disp ( ) ; disp ( Solow model : exogenous growth model ) ; disp ( ) ; %% parameters s = 0. 4 ; % savings r a t e A = 1; % t e c h n o l o g i c a l progress ( Hicks n e u t r a l ) alpha = 0. 3 ; % c a p i t a l share i n production d e l t a = 0. 1 ; % d e p r e c i a t i o n r a t e n = ; % p o p u l a t i o n growth r a t e f p r i n t f ( s A alpha d e l t a n \ n ) ; f p r i n t f ( %6.2 f %6.2 f %6.2 f %6.2 f %6.2 f \ n, s, A, alpha, delta, n ) ; O. Afonso, P. B. Vasconcelos Computational Economics 20 / 27
21 Computational implementation Solution %% steady s t a t e and numerical s o l u t i o n odesolow t, k ) s A k^ alpha ( d e l t a +n ) k ; kss = f s o l v e (@( k ) odesolow ( [ ], k ),10) ; [ t, y ] = my_euler ( odesolow, [ 0, ], 0. 5 kss, ) ; plot ( t, y ) ; i f ~ exist ( OCTAVE_VERSION, b u i l t i n ) % l a b e l s f o r MATLAB xlabel ( $t$, I n t e r p r e t e r, LaTex ) ; ylabel ( $k$, I n t e r p r e t e r, LaTex ) ; else % l a b e l s f o r Octave xlabel ( t ) ; ylabel ( k ) ; end O. Afonso, P. B. Vasconcelos Computational Economics 21 / 27
22 Computational implementation Euler method function [ t, y ] = my_euler ( f, tspan, y0, h ) %( progressive ) Euler method with tspan =[ t0 t f ] and f i x e d % step size to solve the IVP y = f ( t, y ), y ( t0 ) =y0 % i n p u t : % f : f u n c t i o n to i n t e g r a t e ( f =@( t, y ) ) % tspan : i n t e g r a t i o n i n t e r v a l [ t0, t f i n a l ] % y0 : i n i t i a l c o n d i t i o n at t0 % h : ( constant ) step size % output : % t : s p e c i f i c times used % y : s o l u t i o n evaluated at t t0 = tspan ( 1 ) ; t f = tspan ( 2 ) ; t = t0 : h : t f ; y = zeros ( 1, length ( t ) ) ; y ( 1 ) = y0 ; yn = y0 ; tn = t0 ; for n = 1: length ( t ) 1 yn = yn+h f ( tn, yn ) ; y ( n+1) = yn ; tn = tn+h ; end end O. Afonso, P. B. Vasconcelos Computational Economics 22 / 27
23 Numerical results and simulation k t Transition dynamics to steady state O. Afonso, P. B. Vasconcelos Computational Economics 23 / 27
24 Numerical results and simulation Depreciation rate from 0.1 to δ=0.1 δ new = k k (δ new +n)*k (δ+n)*k t k Transition dynamics and Solow diagram: variation on δ O. Afonso, P. B. Vasconcelos Computational Economics 24 / 27
25 Numerical results and simulation Technology from A = 1 to A new = k A=1 A new =1.1 k sak α sa new k α t k Transition dynamics and Solow diagram: variation on A O. Afonso, P. B. Vasconcelos Computational Economics 25 / 27
26 Highlights The Solow model was developed in 1956 by the Nobel Laureate of 1987, Robert Solow. The Solow model shows how an economy changes over time until it gets to steady state. Savings rates determine the level of per capita capital (and output) level in steady state. The higher the savings rate, the higher the level of per capita capital (output) level. Population growth has a negative impact on capital accumulation. The higher the population growth rate, the lower level of capital (output) steady state. However, neither factors can explain sustained economic growth as observed in developed countries. In the current model, per capita capital (output) is invariant in steady state. This chapter introduces the Euler method, the simplest numerical method to solve initial value problems. O. Afonso, P. B. Vasconcelos Computational Economics 26 / 27
27 Main references G. Dahlquist, Å Björck Numerical methods in scientific computing, volume 1. Society for Industrial Mathematics, 2008 E. Süli, D..F. Mayers An introduction to numerical analysis. Cambridge University Press, R. Solow A Contribution to the Theory of Economic Growth. Quarterly Journal of Economics, 70(1): 65 94, O. Afonso, P. B. Vasconcelos Computational Economics 27 / 27
Chapter 12 Ramsey Cass Koopmans model
Chapter 12 Ramsey Cass Koopmans model O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 33 Overview 1 Introduction 2
More informationChapter 10 Skill biased technological change
Chapter 10 Skill biased technological change O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 29 Overview 1 Introduction
More information14.05: Section Handout #1 Solow Model
14.05: Section Handout #1 Solow Model TA: Jose Tessada September 16, 2005 Today we will review the basic elements of the Solow model. Be prepared to ask any questions you may have about the derivation
More informationThe Solow Growth Model
The Solow Growth Model Lectures 5, 6 & 7 Topics in Macroeconomics Topic 2 October 20, 21 & 27, 2008 Lectures 5, 6 & 7 1/37 Topics in Macroeconomics From Growth Accounting to the Solow Model Goal 1: Stylized
More informationThe Solow Growth Model
The Solow Growth Model 1. Set-Up 2. Dynamics, Simulations and Steady-States 3. Comparative Dynamics 4. Golden Rule 5. Convergence 1 Set-Up Closed economy, single good produced each period, Yt. Discrete
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 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 informationEndogenous Growth Theory
Endogenous Growth Theory Lecture Notes for the winter term 2010/2011 Ingrid Ott Tim Deeken October 21st, 2010 CHAIR IN ECONOMIC POLICY KIT University of the State of Baden-Wuerttemberg and National Laboratory
More informationAnalysis of the speed of convergence
Analysis of the speed of convergence Lionel Artige HEC Université de Liège 30 january 2010 Neoclassical Production Function We will assume a production function of the Cobb-Douglas form: F[K(t), L(t),
More informationFrom Difference to Differential Equations I
From Difference to Differential Equations I Start with a simple difference equation x (t + 1) x (t) = g(x (t)). (30) Now consider the following approximation for any t [0, 1], x (t + t) x (t) t g(x (t)),
More informationThe Solow Model. Prof. Lutz Hendricks. January 26, Econ520
The Solow Model Prof. Lutz Hendricks Econ520 January 26, 2017 1 / 28 Issues The production model measures the proximate causes of income gaps. Now we start to look at deep causes. The Solow model answers
More informationSolow Growth Model. Michael Bar. February 28, Introduction Some facts about modern growth Questions... 4
Solow Growth Model Michael Bar February 28, 208 Contents Introduction 2. Some facts about modern growth........................ 3.2 Questions..................................... 4 2 The Solow Model 5
More informationIntermediate Macroeconomics, EC2201. L2: Economic growth II
Intermediate Macroeconomics, EC2201 L2: Economic growth II Anna Seim Department of Economics, Stockholm University Spring 2017 1 / 64 Contents and literature The Solow model. Human capital. The Romer model.
More informationChapter 4 AD AS. O. Afonso, P. B. Vasconcelos. Computational Economics: a concise introduction
Chapter 4 AD AS O. Afonso, P. B. Vasconcelos Computational Economics: a concise introduction O. Afonso, P. B. Vasconcelos Computational Economics 1 / 32 Overview 1 Introduction 2 Economic model 3 Numerical
More informationModeling Economic Growth Using Differential Equations
Modeling Economic Growth Using Differential Equations Chad Tanioka Occidental College February 25, 2016 Chad Tanioka (Occidental College) Modeling Economic Growth using DE February 25, 2016 1 / 28 Overview
More informationEconomic Growth
MIT OpenCourseWare http://ocw.mit.edu 14.452 Economic Growth Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 14.452 Economic Growth: Lecture
More informationFoundations of Modern Macroeconomics Third Edition
Foundations of Modern Macroeconomics Third Edition Chapter 12: Exogenous economic growth Solow-Swan Ben J. Heijdra Department of Economics, Econometrics & Finance University of Groningen 13 December 2016
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 informationGeneric Analysis of Endogenous Growth Models
c November 20, 2017, Christopher D. Carroll Endogenous Generic Analysis of Endogenous Growth Models The neoclassical theory of economic growth, as formulated by Solow (1956), and Cass (1965)-Koopmans (1965),
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 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 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 informationGrowth. Growth Theory. Mark Huggett 1. 1 Georgetown. January 26, 2018
Growth Theory Mark Huggett 1 1 Georgetown January 26, 2018 Growth Theory: The Agenda 1. Facts motivating theory 2. Basic Solow model 3. Model properties 4. How to use the model 5. Full Solow model 6. Use
More informationGrowth Theory: Review
Growth Theory: Review Lecture 1, Endogenous Growth Economic Policy in Development 2, Part 2 March 2009 Lecture 1, Exogenous Growth 1/104 Economic Policy in Development 2, Part 2 Outline Growth Accounting
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 informationEconomic Growth: Lecture 9, Neoclassical Endogenous Growth
14.452 Economic Growth: Lecture 9, Neoclassical Endogenous Growth Daron Acemoglu MIT November 28, 2017. Daron Acemoglu (MIT) Economic Growth Lecture 9 November 28, 2017. 1 / 41 First-Generation Models
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 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 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 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 informationSolow Growth Model. Sang Yoon (Tim) Lee. Jan 9-16, last updated: January 20, Toulouse School of Economics
Solow Growth Model Sang Yoon (Tim) Lee Toulouse School of Economics Jan 9-16, 2018 last updated: January 20, 2018 This Week: Industrialized Countries Kaldor Facts: since we ever measured such things, 1.
More informationLecture notes on modern growth theories
Lecture notes on modern growth theories Part 1 Mario Tirelli Very preliminary material. Not to be circulated without permission of the author. January 18, 2019 Contents 1. Introduction 1 2. Preliminary
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 informationLecture notes on modern growth theories
Lecture notes on modern growth theories Part 1 Mario Tirelli Very preliminary material. Not to be circulated without permission of the author. October 1, 2017 Contents 1. Introduction 1 2. Preliminary
More informationTheoretical premises of the Keynesian approach
origin of Keynesian approach to Growth can be traced back to an article written after the General Theory (1936) Roy Harrod, An Essay in Dynamic Theory, Economic Journal, 1939 Theoretical premises of the
More informationEconomic Growth: Lecture 8, Overlapping Generations
14.452 Economic Growth: Lecture 8, Overlapping Generations Daron Acemoglu MIT November 20, 2018 Daron Acemoglu (MIT) Economic Growth Lecture 8 November 20, 2018 1 / 46 Growth with Overlapping Generations
More informationConvergence behaviour in exogenous growth models
Convergence behaviour in exogenous growth models Jochonia S Mathunjwa and Jonathan R W Temple Department of Economics, University of Bristol 8 Woodland Road, Bristol BS8 TN April 9, 27 Abstract This paper
More informationMacroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models
Macroeconomics II Dynamic macroeconomics Class 1: Introduction and rst models Prof. George McCandless UCEMA Spring 2008 1 Class 1: introduction and rst models What we will do today 1. Organization of course
More informationLecture 2: Intermediate macroeconomics, autumn Lars Calmfors
Lecture 2: Intermediate macroeconomics, autumn 2009 Lars Calmfors 1 Topics Production Labour productivity and economic growth The Solow Model Endogenous growth Long-run effects of the current recession
More informationSolution to Homework 2 - Exogeneous Growth Models
Solution to Homework 2 - Exogeneous Growth Models ECO-3211 Macroeconomia Aplicada (Applied Macroeconomics Question 1: Solow Model with a Fixed Factor 1 The law of motion for capital in the Solow economy
More informationLecture 2: Intermediate macroeconomics, autumn Lars Calmfors
Lecture 2: Intermediate macroeconomics, autumn 2008 Lars Calmfors 1 GDP per capita, percent of OECD average, PPP-adjusted Position 1970 Index Position 1980 Index 1 Switzerland 154 1 USA 140 2 USA 147 2
More informationLecture 1: The Classical Optimal Growth Model
Lecture 1: The Classical Optimal Growth Model This lecture introduces the classical optimal economic growth problem. Solving the problem will require a dynamic optimisation technique: a simple calculus
More informationOne-Sector Models of Endogenous Growth. Instructor: Dmytro Hryshko
One-Sector Models of Endogenous Growth Instructor: Dmytro Hryshko 1 Mid-1980s: dissatisfaction with exogenously driven explanations of long-run productivity growth. 2 It led to construction of models in
More informationMacroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PS 5, preliminary version
Macroeconomics I, UPF Professor ntonio Ciccone SOUTIONS PS 5, preliminary version 1 The Solow K model with transitional dynamics Consider the following Solow economy: production is determined by Y F (K,
More informationDEPARTMENT OF ECONOMICS Fall 2015 P. Gourinchas/D. Romer MIDTERM EXAM
UNIVERSITY OF CALIFORNIA Economics 202A DEPARTMENT OF ECONOMICS Fall 2015 P. Gourinchas/D. Romer MIDTERM EXAM The exam consists of two parts. There are 85 points total. Part I has 18 points and Part II
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 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 informationEquating output per worker to GDP per capita, the growth rate of GDP per capita
3 Homework 3 1. We have seen in class Kaldor s stylized facts of growth in developed countries. The Cobb-Douglas production function is used to replicate fact a. In this exercise, you are asked to show
More information"Solow Model" and Its' Linkage with "Harrod-Domar"
"Solow Model" and Its' Linage with "Harrod-Domar" Harmeet Singh Gulati, Deepinder Kaur 1 SGND KhalsaCollege,Delhi University SGND Khalsa College, Delhi University Abstract: In this paper we aim to lin
More informationECON 5118 Macroeconomic Theory
ECON 5118 Macroeconomic Theory Winter 013 Test 1 February 1, 013 Answer ALL Questions Time Allowed: 1 hour 0 min Attention: Please write your answers on the answer book provided Use the right-side pages
More informationLecture 5: The neoclassical growth model
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 3005 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 5: The neoclassical
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 informationTHE SOLOW-SWAN MODEL WITH A NEGATIVE LABOR GROWTH RATE
Journal of Mathematical Sciences: Advances and Applications Volume 9, Number /,, Pages 9-38 THE SOLOW-SWAN MODEL WITH A NEGATIVE LABOR GROWTH RATE School of Economic Mathematics Southwestern University
More informationECON 402: Advanced Macroeconomics 1. Advanced Macroeconomics, ECON 402. New Growth Theories
ECON 402: Advanced Macroeconomics 1 Advanced Macroeconomics, ECON 402 New Growth Theories The conclusions derived from the growth theories we have considered thus far assumes that economic growth is tied
More informationAdvanced Macroeconomics
Advanced Macroeconomics Endogenous Growth Marcin Kolasa Warsaw School of Economics Marcin Kolasa (WSE) Ad. Macro - Endogenous growth 1 / 18 Introduction The Solow and Ramsey models are exogenous growth
More informationChapter 2. The Solow Growth Model
Chapter 2. The Solow Growth Model Instructor: Dmytro Hryshko 1 / 56 Solow model Solow model is a starting point for more complex models. Abstracts from modeling heterogeneous households (in tastes, abilities,
More informationEconomics 2: Growth (Growth in the Solow Model)
Economics 2: Growth (Growth in the Solow Model) Lecture 3, Week 7 Solow Model - I Definition (Solow Model I) The most basic Solow model with no population growth or technological progress. Solow Model
More informationA Generalized Solow Nonautonomous Model with Delay and Bounded Population Growth
Applied Mathematical Sciences, Vol. 8, 2014, no. 180, 8965-8976 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410817 A Generalized Solow Nonautonomous Model with Delay and Bounded Population
More informationGrowth: Facts and Theories
Notes on Growth: Facts and Theories Intermediate Macroeconomics Spring 2006 Guido Menzio University of Pennsylvania Growth In the last part of the course we are going to study economic growth, i.e. the
More informationNeoclassical Models of Endogenous Growth
Neoclassical Models of Endogenous Growth October 2007 () Endogenous Growth October 2007 1 / 20 Motivation What are the determinants of long run growth? Growth in the "e ectiveness of labour" should depend
More informationDynamical Systems. August 13, 2013
Dynamical Systems Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 Dynamical Systems are systems, described by one or more equations, that evolve over time.
More informationLecture 4 Economic Growth: Foundations
Lecture 4 Economic Growth: Foundations Leopold von Thadden University of Mainz and ECB (on leave) Macroeconomics II, Summer Term 2013 1 / 67 I Motivation This Lecture considers extensions of the basic
More informationDYNAMIC LECTURE 1 UNIVERSITY OF MARYLAND: ECON 600
DYNAMIC LECTURE 1 UNIVERSITY OF MARYLAND: ECON 6 1. differential Equations 1 1.1. Basic Concepts for Univariate Equations. We use differential equations to model situations which treat time as a continuous
More informationEcon 7110 slides Growth models: Solow, Diamond, Malthus. January 8, 2017
Econ 7110 slides Growth models: Solow, Diamond, Malthus January 8, 2017 Production functions Standard setting: Y =output, K (capital) and L (labor) K, L referred to as inputs Usually assumed to satisfy:
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 informationECON 581: Growth with Overlapping Generations. Instructor: Dmytro Hryshko
ECON 581: Growth with Overlapping Generations Instructor: Dmytro Hryshko Readings Acemoglu, Chapter 9. Motivation Neoclassical growth model relies on the representative household. OLG models allow for
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 informationAnother Proof for the Stability of a Modified Solow Model
Applied Mathematical Sciences, Vol 5, 2011, no 25, 1229-1233 Another Proof for the Stability of a Modified Solow Model Massimiliano Ferrara Department SSGES Mediterranean University of Reggio Calabria,
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 informationLecture 15 Real Business Cycle Model. Noah Williams
Lecture 15 Real Business Cycle Model Noah Williams University of Wisconsin - Madison Economics 702/312 Real Business Cycle Model We will have a shock: change in technology. Then we will have a propagation
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 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 informationThe Ramsey Model. (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 2013)
The Ramsey Model (Lecture Note, Advanced Macroeconomics, Thomas Steger, SS 213) 1 Introduction The Ramsey model (or neoclassical growth model) is one of the prototype models in dynamic macroeconomics.
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 informationOn the Dynamic Implications of the Cobb- Douglas Production Function
From the SelectedWorks of Jürgen Antony 2010 On the Dynamic Implications of the Cobb- Douglas Production Function Jürgen Antony, CPB Netherlands Bureau for Economic Policy Analysis Available at: https://works.bepress.com/antony/7/
More informationGeneral motivation behind the augmented Solow model
General motivation behind the augmented Solow model Empirical analysis suggests that the elasticity of output Y with respect to capital implied by the Solow model (α 0.3) is too low to reconcile the model
More information1. Basic Neoclassical Model (Solow Model) (April 14, 2014)
Prof. Dr. Thomas Steger Advanced Macroeconomics I Lecture SS 14 1. Basic Neoclassical Model (Solow Model) (April 14, 2014) Introduction Model setup Intensive production function Capital accumulation The
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 information14.06 Lecture Notes Intermediate Macroeconomics. George-Marios Angeletos MIT Department of Economics
14.06 Lecture Notes Intermediate Macroeconomics George-Marios Angeletos MIT Department of Economics Spring 2004 Chapter 2 The Solow Growth Model (and a look ahead) 2.1 Centralized Dictatorial Allocations
More informationMaster 2 Macro I. Lecture 2 : Balance Growth Paths
2012-2013 Master 2 Macro I Lecture 2 : Balance Growth Paths Franck Portier (based on Gilles Saint-Paul lecture notes) franck.portier@tse-fr.eu Toulouse School of Economics Version 1.1 24/09/2012 Changes
More informationA Summary of Economic Methodology
A Summary of Economic Methodology I. The Methodology of Theoretical Economics All economic analysis begins with theory, based in part on intuitive insights that naturally spring from certain stylized facts,
More informationFoundations of Modern Macroeconomics Third Edition
Foundations of Modern Macroeconomics Third Edition Chapter 16: Overlapping generations in discrete time (sections 16.1 16.2) Ben J. Heijdra Department of Economics, Econometrics & Finance University of
More informationA Contribution to the Empirics of Economic Growth
A Contribution to the Empirics of Economic Growth Albert Alex Zevelev May 6, 2011 1 Intoduction This paper replicates Mankiw, Romer, and Weil s 1992 QJE paper A Contribution to the Empirics of Economic
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 information3 GROWTH AND CAPITAL ACCUMULATION: THE SOLOW MODEL
Economics 314 Coursebook, 2012 Jeffrey Parker 3 GROWTH AND CAPITAL ACCUMULATION: THE SOLOW MODEL Chapter 3 Contents A. Topics and Tools... 1 B. Growth in Continuous Time: Logarithmic and Exponential Functions...
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 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 informationDynamic Optimization: An Introduction
Dynamic Optimization An Introduction M. C. Sunny Wong University of San Francisco University of Houston, June 20, 2014 Outline 1 Background What is Optimization? EITM: The Importance of Optimization 2
More information(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Government Purchases and Endogenous Growth Consider the following endogenous growth model with government purchases (G) in continuous time. Government purchases enhance production, and the production
More informationLecture 4A: The Discrete-Time Overlapping-Generations Model: Basic Theory & Applications
Lecture 4A: The Discrete-Time Overlapping-Generations Model: Basic Theory & Ben J. Heijdra Department of Economics, Econometrics & Finance University of Groningen 13 January 2012 NAKE Dynamic Macroeconomic
More informationMaster 2 Macro I. Lecture 8 : Empirical studies of convergence
2012-2013 Master 2 Macro I Lecture 8 : Empirical studies of convergence Franck Portier (based on Gilles Saint-Paul lecture notes) franck.portier@tse-fr.eu Toulouse School of Economics Version 1.1 14/10/2012
More informationXLVIII Reunión Anual. Noviembre de 2013
ANAES ASOCIACION ARGENTINA DE ECONOMIA POITICA XVIII Reunión Anual Noviembre de 2013 ISSN 1852-0022 ISBN 978-987-28590-1-5 A GENERAIZED VARIABE EASTICITY OF SUBSTITUTION PRODUCTION FUNCTION WITH AN APPICATION
More informationOn Returns to Scale Assumption in Endogenous Growth
International Journal of Sciences: Basic and Applied Research (IJSBAR) ISSN 2307-453 (Print & Online) http://gssrr.org/index.php?journaljournalofbasicandapplied ---------------------------------------------------------------------------------------------------------------------------
More informationFoundation of (virtually) all DSGE models (e.g., RBC model) is Solow growth model
THE BASELINE RBC MODEL: THEORY AND COMPUTATION FEBRUARY, 202 STYLIZED MACRO FACTS Foundation of (virtually all DSGE models (e.g., RBC model is Solow growth model So want/need/desire business-cycle models
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 informationEconomics 202A Suggested Solutions to Problem Set 5
Economics 202A Suggested Solutions to Problem Set 5 David Romer/Galina Hale Spring 1999 1 Romer 3.1. Our R&D model without a capital is Y (t) = A(t)(1 a L )L(t) (1) Ȧ(t) = B[a L L(t)] A(t) θ θ
More informationThe Ramsey/Cass-Koopmans (RCK) Model
c November 2, 217, Christopher D. Carroll RamseyCassKoopmans The Ramsey/Cass-Koopmans (RCK) Model Ramsey (1928), followed much later by Cass (1965) and Koopmans (1965), formulated the canonical model of
More information2 The model with Kaldor-Pasinetti saving
Applied Mathematical Sciences, Vol 6, 2012, no 72, 3569-3573 The Solow-Swan Model with Kaldor-Pasinetti Saving and Time Delay Luca Guerrini Department of Mathematics for Economic and Social Sciences University
More informationToulouse School of Economics, M2 Macroeconomics 1 Professor Franck Portier. Exam Solution
Toulouse School of Economics, 2013-2014 M2 Macroeconomics 1 Professor Franck Portier Exam Solution This is a 3 hours exam. Class slides and any handwritten material are allowed. You must write legibly.
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 information