# Macroeconomics Lecture 1: The Solow Growth Model

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Macroeconomics Lecture 1: The Solow Growth Model Richard G. Pierse 1 Introduction One of the most important long-run issues in macroeconomics is understanding growth. Why do economies grow and what determines how fast they grow? In this lecture we look at a key model of growth developed independently by Robert Solow (1956) and Trevor Swan (1956) and known variously as the Solow growth model, the Solow-Swan growth model or the neo-classical growth model. 2 The Production Function The heart of the theory of growth is a production function relating output to the inputs necessary to create it, namely the factors of production. The production function is written as Y (t) = F (K(t), L(t)) (2.1) where Y is output, K is the input of physical capital (buildings and machines) and L is the input of labour. It will be assumed for simplicity that each worker supplies a single unit of labour so that L also represents the number of workers. The function F () represents the production process that transforms the factors of production into output. Note that since we are considering output of the whole economy, this is an aggregate production function. The simplest assumption that validates the existence of such a production function for the whole economy is that the economy comprises a single good. Alternatively, if we allow for many goods in the economy, we must assume that we can aggregate the production functions for each good into a valid production function for the whole economy. All three variables in (2.1) are functions of time, t, and it is the way in which the factors of production grow over time which will determine the growth of output. Note that Y is a flow variable representing the rate of flow of output while K and L are both stocks of the factors of production, capital and labour, existing at a point 1

2 in time. For convenience, the time argument (t) will henceforth be dropped and the production function rewritten as and Y = F (K, L) (2.2) We make the following assumptions on the derivatives of F (): F/ K = F K > 0 F/ L = F L > 0 (2.3) 2 F/ K 2 = F KK < 0 2 F/ L 2 = F LL < 0 (2.4) 2 F/ K L = F KL > 0. The first two conditions simply state that output is increasing in each of the factor inputs whereas the second two conditions embody the standard assumption of diminishing marginal product. The final condition states that when the input of one factor increases, the marginal product of the other factor increases. Another important assumption that we make is that the production function exhibits constant returns to scale. What this means is that, for any λ > 0, λy = F (λk, λl) (2.5) Constant returns to scale implies that if both the inputs double, then output doubles. It can be justified by the argument of replication, that when capital and labour inputs are doubled, they are used in the same way as before and so output doubles. This implies that the economy is large enough so that the gains from specialisation have been exhausted. The constant returns to scale assumption allows us to work with the production function in intensive form. Setting λ = 1/L we have Y/L = F (K/L, 1) (2.6) or y = f(k) (2.7) where y = Y/L is output per worker (per capita) and k = K/L is capital per worker (the capital-labour ratio). Since f(k) = F (K, L)/L (2.8) and K k = (kl) = L (2.9) k it follows from (2.3) and (2.4) and the chain rule that f k = f K (k) = F K k /L = F K > 0 (2.10) 2

3 and 2 f k = f (k) = f (k) K = F 2 KK k k = F KKL < 0. (2.11) The intensive-form production function is also assumed to satisfy the Inada conditions (Inada (1964)) that: lim f (k) = and lim f (k) = 0. (2.12) k 0 k Intuitively, these conditions state that when k is sufficiently small, its marginal product is very large but that as k becomes large, its marginal product approaches zero. The rôle of these conditions is to ensure that the path of the economy does not diverge. A production function satisfying the conditions (2.10), (2.11) and (2.12) is shown in Figure 1. y f(k) Figure 1: An intensive-form production function: f(k) k 2.1 The Cobb-Douglas Production Function A specific form of production function commonly used in the literature is the Cobb-Douglas production function (Cobb and Douglas (1928)). This takes the 3

4 form: with α > 0 and β > 0. For this production function F (K, L) = K α L β (2.13) F (λk, λl) = (λk) α (λl) β = λ α+β K α L β = λ α+β F (K, L) (2.14) so that, for constant returns to scale, we require that α + β = 1 or β = 1 α. Imposing this condition, the constant returns to scale form of the Cobb-Douglas production function can be written as with 0 < α < 1 or, in intensive form, as F (K, L) = K α L 1 α (2.15) f(k) = F (K/L, 1) = (K/L) α = k α. (2.16) Note that this production function satisfies the conditions (2.10) and (2.11) since and f (k) = αk α 1 > 0 (2.17) f (k) = α(α 1)k α 2 = α(1 α)k α 2 < 0. (2.18) It also can be seen that it satisfies the Inada conditions (2.12). The Cobb-Douglas function is an example of a production function with a constant elasticity of substitution between the factors of production K and L. The elasticity of substitution is a measure of the curvature of the isoquants. For the Cobb-Douglas production function, the slope of the isoquants is Q = F/ K F/ L = and the elasticity is given by [ ] [ 1 Q (L/K).L/K = Q αkα 1 L 1 α (1 α)k α L = α L/K (2.19) α (1 α) α (1 α) ] 1 L/K α L/K = 1 (2.20) (1 α) which is indeed constant. This is illustrated in Figure 2. Points e 1 and e 2 are two different points on the same isoquant K α L 1 α = Y of a Cobb-Douglas production function. The slope of the isoquant at points e 1 and e 2 is given by minus the tangent of the angles A and B respectively. The labour-capital ratio L/K at points e 1 and e 2 is the slope of the chord from the origin to the point, which is the tangent of the angles C and D respectively. Heuristically, the elasticity 4

5 L K α L 1 α = Y e 1 e 2 C D A B Figure 2: Cobb-Douglas isoquant: K α L 1 α = Y K of substitution is the ratio of the change in the angle C to D to the change in the angle A to B. With the Cobb-Douglas production function, this elasticity of substitution is the same at all points on the isoquant and for all isoquants and is equal to unity. With more curved (convex) isoquants, the elasticity of substitution would be lower and with less convex isoquants higher. The Cobb-Douglas is a special member of the class of Constant Elasticity of Substitution (CES) production functions of Arrow et al. (1961) defined by F (K, L) = ( α(bk) ψ + (1 α)[(1 b)l] ψ) 1/ψ (2.21) where 0 < α < 1, 0 < b < 1 and ψ < 1. The elasticity of substitution between capital and labour in the CES production function is given by 1/(1 ψ). It can be shown that the Cobb-Douglas production function is a limiting case of the CES production function where ψ 0. Another limiting case of the CES production function is where ψ so that the elasticity of substitution 1/(1 ψ) 0. In this case the production function reduces to F (K, L) = min(ck, dl) (2.22) 5

6 where c > 0 and d > 0 are constants. This is the Leontief production function in which labour and capital can only be used in fixed proportions d/c so that there can be no substitution between them. An example would be a factory in which each worker operates a single machine. Increasing the number of workers without increasing the number of machines (or vice versa) will not increase output. Output can only be increased by increasing both labour and capital in fixed proportions (in this case one-to-one). Figure 3 illustrates an isoquant of the Leontief production L min(ck, dl) = Y Figure 3: Leontief isoquant: min(ck, dl) = Y K function. The slope of the dotted chord from the origin is the labour-capital ratio d/c and all isoquants of the production function minimise factor inputs along this same chord. 3 A note on growth rates We use the notation ẋ to represent the time derivative of a variable x, dx/dt. The growth rate of x is defined by 1 dx x dt = ẋ x. (3.1) 6

7 Note that if x = y z then, by the rule for the derivative of a product, dx dt = z dy dt + y dz dt (3.2) and, dividing by x, or which, in dot notation, is 1 dx x dt = z dy x dt + y dz x dt 1 dx x dt = 1 dy y dt + 1 dz z dt (3.3) (3.4) ẋ x = ẏ y + ż z. (3.5) This useful result that the growth rate of the product of two variables is equal to the sum of the growth rates of the two variables, generalises to the case of more than two variables. 4 Growth of the factors of production In the production function (2.1), both output Y and the factors of production K and L are assumed to be functions of time. Output changes over time as a result of changes in the factors of production. The capital stock in the economy K is assumed to depreciate at a constant rate δ (0 < δ < 1) as it wears out. Without some capital investment, it would continually decline. However, the capital stock can be increased by investment so that the stock of capital changes over time according to the differential equation K = I δk (4.1) where I is the rate of (gross) capital investment (which is a flow over time). What determines the rate of investment? Total output in the economy, Y, is equal to total expenditure which (in this closed economy assuming no exports or imports and no government) is the sum of consumption expenditure C and investment expenditure I as defined by the expenditure identity Y C + I. (4.2) Output is also equal to total income which is the sum of consumption C and savings S as defined by the income identity Y C + S. (4.3) 7

8 It follows from (4.2) and (4.3) that investment is equal to savings, i.e. I S. (4.4) The savings assumption of the Solow growth model is that savings S are exogenously determined and are assumed to be simply a constant proportion s of income: S = sy (4.5) where 0 < s < 1. Since investment is equal to savings it follows that I = sy = sf (K, L) (4.6) and from (4.1) K = sy δk. (4.7) Finally, it is assumed that the labour force L grows at a constant rate n, 5 The Balanced Growth Path L L = n or L = nl. (4.8) Having made the assumptions (4.5) and (4.8) on the dynamics of the factors of production, we can now derive the key equation of the Solow growth model. Since it follows from the rule for the derivative of a ratio that or, using dot notation, k = K/L (5.1) dk dt = 1 dk L dt K 1 dl L L dt (5.2) k = K L k L L. (5.3) Dividing (4.7) by L we get K L = sy L δ K L (5.4) or K = sy δk L (5.5) and, substituting (5.5) and (4.8) into (5.3), we derive the fundamental growth equation k = sy δk nk (5.6) 8

9 or, using (2.7), k = sf(k) (n + δ)k. (5.7) This equation says that the rate of change of capital per worker is the difference between two terms. The first is investment per worker I/L while the second is break-even investment which is the amount of investment per worker needed just to keep capital per worker constant. y f(k) (n + δ)k sf(k) k Figure 4: The balanced growth path: sf(k ) = (n + δ)k k The second term in (5.7) is linear in k and is increasing as long as (n + δ) > 0 which is a plausible assumption. However, the first term is non-linear since from (2.11) f (k) < 0 and, from the Inada conditions, we know that f (k) decreases from at k = 0 to 0 at k =. It follows that when k is small, the first term will be larger than the second term and so k will be positive and k will be increasing. However, as k increases f (k) decreases and there will come a point when the second term outweighs the first and beyond this k will become negative so that k starts to decrease. The equilibrium point is where k = 0. Let us denote this value of k as k. For k > 0, this equilibrium is a unique one and, as long as (n + δ) > 0, it is a stable equilibrium since, if k is above or below k, k will adjust to bring it 9

10 back to k. The equilibrium is illustrated in Figure 4. The equilibrium point defines the balanced growth path. At this point k will be constant at k but the capital stock itself will, (using 3.5), be growing at the rate K = Lk (5.8) K K = L L + k k = n. (5.9) Similarly, although in equilibrium output per worker will be constant at f(k ), output itself, Y = Lf(k ), (5.10) will also be growing at the rate n. 6 Savings and the Golden Rule of Accumulation How can a country attempt to change its rate of growth? One possibility is to alter the savings rate s. Increasing the savings rate will lead to a move to a new steady state with higher output per worker y. However, the effect on consumption per worker c = C/L is not clear. Dividing the income identity (4.3) by Y we have or and, substituting into (5.7), Y Y = C Y + S Y s = 1 C Y (6.1) (6.2) k = (1 C )f(k) (n + δ)k = 0 (6.3) Y in steady state. Rearranging, and using to denote steady state values but so we have C Y f(k ) = f(k ) (n + δ)k (6.4) C Y f(k ) = C Y Y L = c (6.5) c (s) = f(k (s)) (n + δ)k (s) (6.6) 10

11 y f(k) y f (k) = n + δ (n + δ)k s sf(k) k Figure 5: The golden rate of savings: f (k (s)) = n + δ k where we have made explicit the dependency of the steady state capital per worker k and the steady state consumption per worker c on the savings rate s. This equation is known as the golden rule of capital accumulation. It defines the relationship between the steady-state consumption per worker c and the savings rate s and this relationship is non-linear. For small values of k, c is increasing in k but for high values c is decreasing in k. c is maximised by setting dc/ds to zero which implies (f (k (s)) (n + δ)) dk = 0. (6.7) ds Since dk /ds > 0, this implies f (k (s)) = n + δ. (6.8) The savings rate, s, that satisfies this equation and maximises consumption per worker is known as the golden rate of savings. It is illustrated in Figure 5. At the equilbrium point the slope of the production function f(k) is equal to the slope of the line (n + δ)k. The golden rate of savings is s and consumption per head c is equal to the distance y minus s. 11

13 References [1] Arrow, K. J., H. B. Chenery, B. S. Minhas and R. M. Solow (1961), Capitallabor substitution and economic efficiency, Review of Economics and Statistics, 43, [2] Cobb, C. W. and P. H. Douglas (1928), A theory of production, American Economic Review, 18, [3] Inada, K-I. (1964), Some structural characteristics of turnpike theorems, Review of Economic Studies, 31, [4] Solow, R. F. (1956), A contribution to the theory of economic growth, Quarterly Journal of Economics, 70, [5] Swan, T. W. (1956), Economic growth and capital accumulation, Economic Record, 32,

### MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT

MASTER IN ENGINEERING AND TECHNOLOGY MANAGEMENT ECONOMICS OF GROWTH AND INNOVATION Lecture 1, January 23, 2004 Theories of Economic Growth 1. Introduction 2. Exogenous Growth The Solow Model Mandatory

### Chapters 7 and 8 Solow Growth Model Basics

Chapters 7 and 8 Solow Growth Model Basics The Solow growth model breaks the growth of economies down into basics. It starts with our production function Y = F (K, L) and puts in per-worker terms. Y L

### Review Questions - CHAPTER 8

Review Questions - CHAPTER 8 1. The formula for steady-state consumption per worker (c*) as a function of output per worker and investment per worker is: A) c* = f(k*) δk*. B) c* = f(k*) + δk*. C) c* =

### The Golden Rule. Where investment I is equal to the savings rate s times total production Y: So consumption per worker C/L is equal to:

The Golden Rule Choosing a National Savings Rate What can we say about economic policy and long-run growth? To keep matters simple, let us assume that the government can by proper fiscal and monetary policies

### 14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model

14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model Daron Acemoglu MIT November 1 and 3, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 2 and 3 November 1 and 3, 2011. 1 / 96 Solow Growth

### Economic Growth: Lecture 2: The Solow Growth Model

14.452 Economic Growth: Lecture 2: The Solow Growth Model Daron Acemoglu MIT October 29, 2009. Daron Acemoglu (MIT) Economic Growth Lecture 2 October 29, 2009. 1 / 68 Transitional Dynamics in the Discrete

### k = sf(k) (δ + n + g)k = 0. sy (δ + n + g)y 2 = 0. Solving this, we find the steady-state value of y:

CHAPTER 8 Economic Growth II Questions for Review 1. In the Solow model, we find that only technological progress can affect the steady-state rate of growth in income per worker. Growth in the capital

### Chapter 7: Economic Growth part 1

Chapter 7: Economic Growth part 1 Learn the closed economy Solow model See how a country s standard of living depends on its saving and population growth rates Learn how to use the Golden Rule to find

### The Solow Model. Savings and Leakages from Per Capita Capital. (n+d)k. sk^alpha. k*: steady state 0 1 2.22 3 4. Per Capita Capital, k

Savings and Leakages from Per Capita Capital 0.1.2.3.4.5 The Solow Model (n+d)k sk^alpha k*: steady state 0 1 2.22 3 4 Per Capita Capital, k Pop. growth and depreciation Savings In the diagram... sy =

### Calibration of Normalised CES Production Functions in Dynamic Models

Discussion Paper No. 06-078 Calibration of Normalised CES Production Functions in Dynamic Models Rainer Klump and Marianne Saam Discussion Paper No. 06-078 Calibration of Normalised CES Production Functions

### ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

### Lecture 14 More on Real Business Cycles. Noah Williams

Lecture 14 More on Real Business Cycles Noah Williams University of Wisconsin - Madison Economics 312 Optimality Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1)

### 8. Average product reaches a maximum when labor equals A) 100 B) 200 C) 300 D) 400

Ch. 6 1. The production function represents A) the quantity of inputs necessary to produce a given level of output. B) the various recipes for producing a given level of output. C) the minimum amounts

### Economic Growth. (c) Copyright 1999 by Douglas H. Joines 1

Economic Growth (c) Copyright 1999 by Douglas H. Joines 1 Module Objectives Know what determines the growth rates of aggregate and per capita GDP Distinguish factors that affect the economy s growth rate

### Preparation course MSc Business&Econonomics: Economic Growth

Preparation course MSc Business&Econonomics: Economic Growth Tom-Reiel Heggedal Economics Department 2014 TRH (Institute) Solow model 2014 1 / 27 Theory and models Objective of this lecture: learn Solow

Duration: 120 min INTRODUCTION TO ADVANCED MACROECONOMICS Preliminary Exam with answers September 2014 Format of the mock examination Section A. Multiple Choice Questions (20 % of the total marks) Section

### Productioin OVERVIEW. WSG5 7/7/03 4:35 PM Page 63. Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.

WSG5 7/7/03 4:35 PM Page 63 5 Productioin OVERVIEW This chapter reviews the general problem of transforming productive resources in goods and services for sale in the market. A production function is the

### Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti)

Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Kjetil Storesletten September 10, 2013 Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44 Growth

### Lecture 2 Dynamic Equilibrium Models : Finite Periods

Lecture 2 Dynamic Equilibrium Models : Finite Periods 1. Introduction In macroeconomics, we study the behavior of economy-wide aggregates e.g. GDP, savings, investment, employment and so on - and their

### Economic Growth: Theory and Empirics (2012) Problem set I

Economic Growth: Theory and Empirics (2012) Problem set I Due date: April 27, 2012 Problem 1 Consider a Solow model with given saving/investment rate s. Assume: Y t = K α t (A tl t ) 1 α 2) a constant

### Name: Date: 3. Variables that a model tries to explain are called: A. endogenous. B. exogenous. C. market clearing. D. fixed.

Name: Date: 1 A measure of how fast prices are rising is called the: A growth rate of real GDP B inflation rate C unemployment rate D market-clearing rate 2 Compared with a recession, real GDP during a

### The Real Business Cycle Model

The Real Business Cycle Model Ester Faia Goethe University Frankfurt Nov 2015 Ester Faia (Goethe University Frankfurt) RBC Nov 2015 1 / 27 Introduction The RBC model explains the co-movements in the uctuations

### Economic Growth I: Capital Accumulation and Population Growth

CHAPTER 8 : Capital Accumulation and Population Growth Modified for ECON 2204 by Bob Murphy 2016 Worth Publishers, all rights reserved IN THIS CHAPTER, YOU WILL LEARN: the closed economy Solow model how

### The Aggregate Production Function Revised: January 9, 2008

Global Economy Chris Edmond The Aggregate Production Function Revised: January 9, 2008 Economic systems transform inputs labor, capital, raw materials into products. We use a theoretical construct called

### 1. Theories of Economic Growth

0DVWHULQ(QJLQHHULQJDQG7HFKQRORJ\ 0DQDJHPHQW (&2120,&62)*52:7+\$1',1129\$7,21 /HFWXUH 1. Theories of Economic Growth 1.1- Introduction 1.2- Exogenous Growth 1.2.1- The Solow Model 5HDGLQJV Mandatory: Romer,

### The previous chapter introduced a number of basic facts and posed the main questions

2 The Solow Growth Model The previous chapter introduced a number of basic facts and posed the main questions concerning the sources of economic growth over time and the causes of differences in economic

### Graduate Macro Theory II: Notes on Investment

Graduate Macro Theory II: Notes on Investment Eric Sims University of Notre Dame Spring 2011 1 Introduction These notes introduce and discuss modern theories of firm investment. While much of this is done

### VI. Real Business Cycles Models

VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized

### MA Macroeconomics 10. Growth Accounting

MA Macroeconomics 10. Growth Accounting Karl Whelan School of Economics, UCD Autumn 2014 Karl Whelan (UCD) Growth Accounting Autumn 2014 1 / 20 Growth Accounting The final part of this course will focus

### Does Debt Relief enhance Growth in Third World Countries?

Does Debt Relief enhance Growth in Third World Countries? Catherine Isabelle Cax Department of Economics, University of Copenhagen Supervisor: Carl-Johan Dalgaard Opponent: Abdulaziz Shifa 24th of November

### Sample Midterm Solutions

Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

### REVIEW OF MICROECONOMICS

ECO 352 Spring 2010 Precepts Weeks 1, 2 Feb. 1, 8 REVIEW OF MICROECONOMICS Concepts to be reviewed Budget constraint: graphical and algebraic representation Preferences, indifference curves. Utility function

### Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

### 14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth

14.452 Economic Growth: Lectures 6 and 7, Neoclassical Growth Daron Acemoglu MIT November 15 and 17, 211. Daron Acemoglu (MIT) Economic Growth Lectures 6 and 7 November 15 and 17, 211. 1 / 71 Introduction

### Lecture 3: The Theory of the Banking Firm and Banking Competition

Lecture 3: The Theory of the Banking Firm and Banking Competition This lecture focuses on the industrial organisation approach to the economics of banking, which considers how banks as firms react optimally

### Agenda. Productivity, Output, and Employment, Part 1. The Production Function. The Production Function. The Production Function. The Demand for Labor

Agenda Productivity, Output, and Employment, Part 1 3-1 3-2 A production function shows how businesses transform factors of production into output of goods and services through the applications of technology.

### CHAPTER 7 Economic Growth I

CHAPTER 7 Economic Growth I Questions for Review 1. In the Solow growth model, a high saving rate leads to a large steady-state capital stock and a high level of steady-state output. A low saving rate

### TAXATION AND SAVINGS. Robin Boadway Queen s University Kingston, Canada. Day Six

TAXATION AND SAVINGS by Robin Boadway Queen s University Kingston, Canada April, 2004 Day Six Overview TAXATION AND SAVINGS We will first summarize the two-period life cycle savings model and derive some

### INVESTMENT DECISIONS and PROFIT MAXIMIZATION

Lecture 6 Investment Decisions The Digital Economist Investment is the act of acquiring income-producing assets, known as physical capital, either as additions to existing assets or to replace assets that

Graduate Macro Theory II: The Real Business Cycle Model Eric Sims University of Notre Dame Spring 2011 1 Introduction This note describes the canonical real business cycle model. A couple of classic references

### 19 : Theory of Production

19 : Theory of Production 1 Recap from last session Long Run Production Analysis Return to Scale Isoquants, Isocost Choice of input combination Expansion path Economic Region of Production Session Outline

### University of Saskatchewan Department of Economics Economics 414.3 Homework #1

Homework #1 1. In 1900 GDP per capita in Japan (measured in 2000 dollars) was \$1,433. In 2000 it was \$26,375. (a) Calculate the growth rate of income per capita in Japan over this century. (b) Now suppose

### UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON4310 Intertemporal macroeconomics Date of exam: Thursday, November 27, 2008 Grades are given: December 19, 2008 Time for exam: 09:00 a.m. 12:00 noon

### Universidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model

Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The Ramsey-Cass-Koopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous

### Multi-variable Calculus and Optimization

Multi-variable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multi-variable Calculus and Optimization 1 / 51 EC2040 Topic 3 - Multi-variable Calculus

### Problem 1. Steady state values for two countries with different savings rates and population growth rates.

Mankiw, Chapter 8. Economic Growth II: Technology, Empirics and Policy Problem 1. Steady state values for two countries with different savings rates and population growth rates. To make the problem more

### Long Run Economic Growth Agenda. Long-run Economic Growth. Long-run Growth Model. Long-run Economic Growth. Determinants of Long-run Growth

Long Run Economic Growth Agenda Long-run economic growth. Determinants of long-run growth. Production functions. Long-run Economic Growth Output is measured by real GDP per capita. This measures our (material)

### . In this case the leakage effect of tax increases is mitigated because some of the reduction in disposable income would have otherwise been saved.

Chapter 4 Review Questions. Explain how an increase in government spending and an equal increase in lump sum taxes can generate an increase in equilibrium output. Under what conditions will a balanced

### Chapter 4 Technological Progress and Economic Growth

Chapter 4 Technological Progress and Economic Growth 4.1 Introduction Technical progress is defined as new, and better ways of doing things, and new techniques for using scarce resources more productively.

### Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey

Towards a Structuralist Interpretation of Saving, Investment and Current Account in Turkey MURAT ÜNGÖR Central Bank of the Republic of Turkey http://www.muratungor.com/ April 2012 We live in the age of

### This paper is not to be removed from the Examination Halls

This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON EC2065 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas

### Note on growth and growth accounting

CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis

### Prep. Course Macroeconomics

Prep. Course Macroeconomics Intertemporal consumption and saving decision; Ramsey model Tom-Reiel Heggedal tom-reiel.heggedal@bi.no BI 2014 Heggedal (BI) Savings & Ramsey 2014 1 / 30 Overview this lecture

Phd Macro, 2007 (Karl Whelan) 1 Real Business Cycle Models The Real Business Cycle (RBC) model introduced in a famous 1982 paper by Finn Kydland and Edward Prescott is the original DSGE model. 1 The early

### 1 National Income and Product Accounts

Espen Henriksen econ249 UCSB 1 National Income and Product Accounts 11 Gross Domestic Product (GDP) Can be measured in three different but equivalent ways: 1 Production Approach 2 Expenditure Approach

### Constrained optimization.

ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

### Agenda. Long-Run Economic Growth, Part 1. The Sources of Economic Growth. Long-Run Economic Growth. The Sources of Economic Growth

Agenda The Sources of Economic Growth Long-Run Economic Growth, Part 1 Growth Dynamics: 8-1 8-2 Long-Run Economic Growth Countries have grown at very different rates over long spans of time. The Sources

### 2007/8. The problem of non-renewable energy resources in the production of physical capital. Agustin Pérez-Barahona

2007/8 The problem of non-renewable energy resources in the production of physical capital Agustin Pérez-Barahona CORE DISCUSSION PAPER 2007/8 The problem of non-renewable energy resources in the production

### WORKING PAPER SERIES 09-2013. Government Debt, the Real Interest Rate and External Balance in an Endogenous Growth Model of a Small Open Economy

ATHENS UNIVERSITY OF ECONOMICS AND BUSINESS DEPARTMENT OF ECONOMICS WORKING PAPER SERIES 09-2013 Government Debt, the Real Interest Rate and External Balance in an Endogenous Growth Model of a Small Open

### Profit Maximization. PowerPoint Slides prepared by: Andreea CHIRITESCU Eastern Illinois University

Profit Maximization PowerPoint Slides prepared by: Andreea CHIRITESCU Eastern Illinois University 1 The Nature and Behavior of Firms A firm An association of individuals Firms Who have organized themselves

Real Business Cycle Models Lecture 2 Nicola Viegi April 2015 Basic RBC Model Claim: Stochastic General Equlibrium Model Is Enough to Explain The Business cycle Behaviour of the Economy Money is of little

### Neoclassical growth theory

Chapter 1 Neoclassical growth theory 1.1 The Solow growth model The general questions of growth: What are the determinants of long-run economic growth? How can we explain the vast differences in both output

### The Real Business Cycle model

The Real Business Cycle model Spring 2013 1 Historical introduction Modern business cycle theory really got started with Great Depression Keynes: The General Theory of Employment, Interest and Money Keynesian

### Endogenous Growth Theory

Chapter 3 Endogenous Growth Theory 3.1 One-Sector Endogenous Growth Models 3.2 Two-sector Endogenous Growth Model 3.3 Technological Change: Horizontal Innovations References: Aghion, P./ Howitt, P. (1992),

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### Economic Growth. Chapter 11

Chapter 11 Economic Growth This chapter examines the determinants of economic growth. A startling fact about economic growth is the large variation in the growth experience of different countries in recent

### Preparation course Msc Business & Econonomics

Preparation course Msc Business & Econonomics The simple Keynesian model Tom-Reiel Heggedal BI August 2014 TRH (BI) Keynes model August 2014 1 / 19 Assumptions Keynes model Outline for this lecture: Go

### ECON 5110 Class Notes Overview of New Keynesian Economics

ECON 5110 Class Notes Overview of New Keynesian Economics 1 Introduction The primary distinction between Keynesian and classical macroeconomics is the flexibility of prices and wages. In classical models

### GDP: The market value of final goods and services, newly produced WITHIN a nation during a fixed period.

GDP: The market value of final goods and services, newly produced WITHIN a nation during a fixed period. Value added: Value of output (market value) purchased inputs (e.g. intermediate goods) GDP is a

### Notes on the Theories of Growth

Notes on the Theories of Growth Economic Growth & Development These recitation notes cover basic concepts in economic growth theory relevant to the cases seen in class (Singapore, Indonesia, Japan, and

### Outline of model. Factors of production 1/23/2013. The production function: Y = F(K,L) ECON 3010 Intermediate Macroeconomics

ECON 3010 Intermediate Macroeconomics Chapter 3 National Income: Where It Comes From and Where It Goes Outline of model A closed economy, market-clearing model Supply side factors of production determination

### Long Run Growth Solow s Neoclassical Growth Model

Long Run Growth Solow s Neoclassical Growth Model 1 Simple Growth Facts Growth in real GDP per capita is non trivial, but only really since Industrial Revolution Dispersion in real GDP per capita across

### Chapter 3 A Classical Economic Model

Chapter 3 A Classical Economic Model what determines the economy s total output/income how the prices of the factors of production are determined how total income is distributed what determines the demand

### Endogenous Growth Models

Endogenous Growth Models Lorenza Rossi Goethe University 2011-2012 Endogenous Growth Theory Neoclassical Exogenous Growth Models technological progress is the engine of growth technological improvements

### The Optimal Growth Problem

LECTURE NOTES ON ECONOMIC GROWTH The Optimal Growth Problem Todd Keister Centro de Investigación Económica, ITAM keister@itam.mx January 2005 These notes provide an introduction to the study of optimal

### ECON 5010 Class Notes Endogenous Growth Theory

ECON 5010 Class Notes Endogenous Growth Theory 1 Introduction One drawbac of the Solow model is that long-run growth in per capita income is entirely exogenous. In the absence of exogenous technological

### Endogenous Growth Theory

Endogenous Growth Theory Motivation The Solow and Ramsey models o er valuable insights but have important limitations: Di erences in capital accummulation cannot satisfactorily account for the prevailing

### The RBC methodology also comes down to two principles:

Chapter 5 Real business cycles 5.1 Real business cycles The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory

Graduate Macroeconomics 2 Lecture 1 - Introduction to Real Business Cycles Zsófia L. Bárány Sciences Po 2014 January About the course I. 2-hour lecture every week, Tuesdays from 10:15-12:15 2 big topics

### Macroeconomic Effects of Financial Shocks Online Appendix

Macroeconomic Effects of Financial Shocks Online Appendix By Urban Jermann and Vincenzo Quadrini Data sources Financial data is from the Flow of Funds Accounts of the Federal Reserve Board. We report the

### Money and Public Finance

Money and Public Finance By Mr. Letlet August 1 In this anxious market environment, people lose their rationality with some even spreading false information to create trading opportunities. The tales about

### Lecture 1: The intertemporal approach to the current account

Lecture 1: The intertemporal approach to the current account Open economy macroeconomics, Fall 2006 Ida Wolden Bache August 22, 2006 Intertemporal trade and the current account What determines when countries

### Keywords: Overlapping Generations Model, Tax Reform, Turkey

SIMULATING THE TURKISH TAX SYSTEM ADEM İLERİ Middle East Technical University Department of Economics aileri@metu.edu.tr PINAR DERİN-GÜRE Middle East Technical University Department of Economics pderin@metu.edu.tr

### The Budget Deficit, Public Debt and Endogenous Growth

The Budget Deficit, Public Debt and Endogenous Growth Michael Bräuninger October 2002 Abstract This paper analyzes the effects of public debt on endogenous growth in an overlapping generations model. The

### Name: Final Exam Econ 219 Spring You can skip one multiple choice question. Indicate clearly which one

Name: Final Exam Econ 219 Spring 2005 This is a closed book exam. You are required to abide all the rules of the Student Conduct Code of the University of Connecticut. You can skip one multiple choice

### Lecture notes for Choice Under Uncertainty

Lecture notes for Choice Under Uncertainty 1. Introduction In this lecture we examine the theory of decision-making under uncertainty and its application to the demand for insurance. The undergraduate

### Chapter 13 Real Business Cycle Theory

Chapter 13 Real Business Cycle Theory Real Business Cycle (RBC) Theory is the other dominant strand of thought in modern macroeconomics. For the most part, RBC theory has held much less sway amongst policy-makers

### TOTAL FACTOR PRODUCTVITY IN STEEL COMPANIES TECHNIQUE OF MEASUREMENT:-

TOTAL FACTOR PRODUCTVITY IN STEEL COMPANIES TECHNIQUE OF MEASUREMENT:- Generally, the total factor productivity compares total output to a weighted composition of inputs, usually labour and capital. 1

Advanced Macroeconomics (ECON 402) Lecture 8 Real Business Cycle Theory Teng Wah Leo Some Stylized Facts Regarding Economic Fluctuations Having now understood various growth models, we will now delve into

### EXOGENOUS GROWTH MODELS

EXOGENOUS GROWTH MODELS Lorenza Rossi Goethe University 2011-2012 Course Outline FIRST PART - GROWTH THEORIES Exogenous Growth The Solow Model The Ramsey model and the Golden Rule Introduction to Endogenous

MA Advanced Macroeconomics: 7. The Real Business Cycle Model Karl Whelan School of Economics, UCD Spring 2015 Karl Whelan (UCD) Real Business Cycles Spring 2015 1 / 38 Working Through A DSGE Model We have

### 14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth

14.452 Economic Growth: Lecture 11, Technology Diffusion, Trade and World Growth Daron Acemoglu MIT December 2, 2014. Daron Acemoglu (MIT) Economic Growth Lecture 11 December 2, 2014. 1 / 43 Introduction

### MACROECONOMICS SECTION

MACROECONOMICS SECTION GENERAL TIPS Be sure every graph is carefully labeled and explained. Every answer must include a section that contains a response to WHY the result holds. Good resources include

### Environmental problems and economic development in an endogenous fertility model

University of Heidelberg Department of Economics Discussion Paper Series No. 428 Environmental problems and economic development in an endogenous fertility model Frank Jöst, Martin Quaas and Johannes Schiller

### Problem Set #3 Answer Key

Problem Set #3 Answer Key Economics 305: Macroeconomic Theory Spring 2007 1 Chapter 4, Problem #2 a) To specify an indifference curve, we hold utility constant at ū. Next, rearrange in the form: C = ū

### Macroeconomics 2. Technological progress and growth: The general Solow model. Mirko Wiederholt. Goethe University Frankfurt.

Macroeconomics 2 Technological progress and growth: The general Solow model Mirko Wiederholt Goethe University Frankfurt Lecture 3 irko Wiederholt (Goethe University Frankfurt) Macroeconomics 2 Lecture

### Keynesian Macroeconomic Theory

2 Keynesian Macroeconomic Theory 2.1. The Keynesian Consumption Function 2.2. The Complete Keynesian Model 2.3. The Keynesian-Cross Model 2.4. The IS-LM Model 2.5. The Keynesian AD-AS Model 2.6. Conclusion

### = C + I + G + NX ECON 302. Lecture 4: Aggregate Expenditures/Keynesian Model: Equilibrium in the Goods Market/Loanable Funds Market

Intermediate Macroeconomics Lecture 4: Introduction to the Goods Market Review of the Aggregate Expenditures model and the Keynesian Cross ECON 302 Professor Yamin Ahmad Components of Aggregate Demand