# Note on growth and growth accounting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 of growth are set out. Notation used in growth theory. Use is often made of the so-called dot notation, to refer to the rate of change of key variables. dot is the rate of change of output and is closely related to the familiar concept of the change in output between periodand period,, which is often abbreviated as. stands for the continuous time equivalent of and is defined as the time derivative of,that is. To summarize: (Rate of change, discrete time) (Rate of change, continuous time) Does it matter whether we work in continuous time or discrete time? Generally speaking in growth theory it is easier and requires less cumbersome notation to derive the results in continuous time. 1 The (proportional) rate of growth is defined in discrete time as and in continuous time as (Growth rate, discrete time) (Growth rate, continuous time) In each case, the growth rate, e.g. 0.02, is multiplied by to produce a percentage growth rate. Calculating growth rates. As this chapter is about economic growth it is crucial to understand the concept of a growth rate and how to measure it using data. The simplest notion of a growth rate is the percentage change in GDP between periods (quarters or years) as defined above. For example US GDP in the year 2000 was bill. and it was bill. in the year 1999 (both measured in 1995 prices). Hence the US economy grew by 4.2% between 1999 and Once we move beyond growth rates over very short periods, it is necessary to calculate the socalled compound growth rate. Economists interested in growth often focus on average annual growth rates over lengthy periods of time. We take an example and compare three methods. First, we calculate the average growth rate in Chinese GDP per capita between 1988 and 1998 using a long average method: p.a.. Second, we calculate the annual growth rate each year of the decade and take the average: the answer is p.a.. Third, we calculate the annual compound growth rate using a formula: AGR (annual percentage growth rate) = p.a.. Note that it is the function on the calculator that must be used since this is the natural logarithm function although we use in the text, this always refers to the natural logarithm. 1 From a mathematical point of view we could prove all our results in discrete time as well, and the equations we have derived in continuous time would hold as approximations to the discrete time results. 1

2 2 0. NOTE ON GROWTH AND GROWTH ACCOUNTING The long average method is incorrect: it overstates the growth rate because it neglects the fact that the base for growth is continuously rising. 2 The average of the annual growth rates will be a reasonable approximation to the compound growth rate for low growth rates but loses accuracy as the growth rate rises. The comparison between the average of the annual growth rates and the compound growth rate is exactly the same as between simple interest (compounded annually) and compound interest (compounded continuously). The AGR is the equivalent of the APR for interest rates (the annual percentage rate of interest reported on credit card bills, for example). We need to explain some more concepts before we return to show where the formula for the annual compound growth rate (AGR) comes from Growth rates, exponential and log functions. To see that the growth of GDP in discrete time, is the same as the growth rate in continuous time,, when the time period is short enough, let us start from the proportional growth of GDP between two periods in time given by,and, where is small. When we make this interval smaller and smaller, we say that we are taking the limit as tends to zero: (1.2) We express the proportional growth expression per unit of time and then the rules of calculus tell us that if we make this period of time very small the first term in the brackets is nothing other than the definition of the derivative of output with respect to time,. is known as the instantaneous growth rate. Thus when we quote, this means that the growth rate of GDP during the period of interest is 2%. It is very useful to see that the growth rate can be expressed in several equivalent ways: (Equivalent expressions, growth rate) In order to see that the last equality holds we can use the chain rule of calculus and the fact that the derivative of the log of a variable,,is, i.e. : (1.3) We now explore further what economists mean when they say that output grows exponentially. Let us start with the previous example where for some economy. In this case, if we know the level of output initially,, and the growth rate, then the level of output at timecomes from the equation, where is the exponential function and the growth rate is assumed to be constant. How do we know that this equation describes the evolution of output for our economy that grows at rate? It is easier to see this if we do the proof in reverse. That is, let us start from the equation: (1.1) (1.4) where stands for the initial value of output. Taking logs of both the left and the right side of this we obtain: (1.5) If we now take derivatives with respect to time of this expression we obtain 2 To see why the first method is incorrect, apply the 7.16% annual growth rate to the base year GDP per capita of \$1816 and then for each subsequent year. The result is a level of GDP p.c. in 1998 far higher than that recorded (\$3117).

3 1. GROWTH AND THE GROWTH RATE Normal Scale 10% 5 Log scale 10% % % 5% % FIGURE 1. Simulation of exponential growth on a normal and logarithmic scale. (1.6) (1.7) The initial value of output is of course fixed at all future values and hence. Thus we have shown that, as expected. An understanding of the relationship between exponential growth and logs is very useful for any analysis of growth. 3 As the plots in the left hand panel of Fig.1 show, once we have the initial value for GDP and the growth rate, we can read off the level of output at any subsequent date. Growth paths for GDP are shown for a 2%, 5% and 10% growth rate. We also note that the equation for is linear, with the slope of the line equal to the growth rate. The growth paths for GDP are shown in the right hand panel using the log scale: the growth paths are straight lines. A second reason for understanding the relationship between logs and growth rates is that this relationship lies behind some very useful rules for handling growth rates. The following rules are used frequently: (1) if then. This is useful because it allows us to go from levels to growth rates. For example, if we begin with the widely used production function, the Cobb Douglas, (Cobb Douglas production function) where and first take logs: then differentiate with respect to time and use the fact that we get: We can see from this that the growth rate of output is a weighted average of the growth rates of capital and labour inputs. 3 For a concise and insightful discussion of the exponential and logarithmic functions, see Chapter 9 in M. Pemberton and N. Rau Mathematics for Economists. Manchester University Press, 2001.

4 4 0. NOTE ON GROWTH AND GROWTH ACCOUNTING (2) the growth rate of the ratio of two variables is equal to the difference between the two growth rates:. This rule is used often. For example, it says that the growth rate of output per worker ( ) is equal to the growth rate of output ( of workers ( ). To show this, the same technique is used as for (1): and hence. ) minus the growth rate (3) the growth rate of the product of two variables is equal to the sum of their growth rates:. This follows the same logic as for Useful definitions for growth theory. In growth models in continuous time, the concept of the growth rate is, which we can express also using and equivalently as. As noted above, the growth rate is the instantaneous growth rate. This may be calculated from data as follows: whereis the number of years, is the base year level of output and the final year level. is multiplied by 100 to get the percentage growth rate. This is often referred to as the log difference method of calculating the growth rate. When referring to the growth of output, economists typically mean the annual growth rate (the discrete time concept), rather than the instantaneous growth rate (the continuous time concept). Similarly, when discussing interest rates, it is the annual percentage rate (APR) rather than the instantaneous rate that is typically calculated. The annual percentage growth rate (or annual compound growth rate as it is sometimes called) is expressed as a percentage and this is the formula for calculating the compound growth rate provided above: The relationship between the different concepts of the growth rate is clarified by noting that when we consider growth for one year, the familiar proportional rate of growth is(multiplied by 100): Extending to many periods, we have This illustrates that the is the multi-period analogue to the one-period proportional growth rate.

5 2. GROWTH ACCOUNTING Summary. When using data to measure economic growth, the average of the annual percentage growth rates is a close approximation to the true annual (compound) percentage growth rate () when the growth rate is low (less than 5% p.a.). In the earlier example of Chinese growth, the average of the annual percentage growth rates is p.a. and the is p.a.. Taking the difference between the natural logarithms of output at the start of the period and the end of the period and dividing by the length of the period gives the instantaneous growth rate,. When multiplied by 100, this is the percentage growth rate in continuous time. To get the, we use the fact that. This is the percentage growth rate in discrete time. When referring to data, economists normally use discrete time concepts and therefore refer to the however growth theory is normally expressed in continuous time, with reference to. When growth rates are low, these are close to each other. In the example of Chinese growth, the p.a. and p.a.. Growth theory makes heavy use of the following equivalent expressions for the (instantaneous) growth rate: and. Frequent use is made of the procedure of taking logs and differentiating to go from expressions in levels to growth rates, and of the rules and. 2. Growth accounting The idea of growth account is to account for the contribution to the growth of output made by the growth of factor inputs (capital and labour) and to associate any growth unaccounted for to technological progress. Solow referred to this residual as total factor productivity growth. Total factor productivity growth captures the impact of intangible aspects of human progress that allow both labour and capital to increase their productivity. Thinking of the period 1990 to 2000, what has made the US economy more productive and has increased welfare was not simply the fact that firms have invested and bought computers, but rather that the information revolution has allowed new machines based on computer technology to be more productive and workers who use computers to be more efficient. Solow s method of calculating total factor productivity growth is known as (Solow) growth accounting. To see how this works we can start from a production function defined in terms of capital, labour and an index of the level of technological progress given by, which is a function of time: (2.1) Let us now take logs of the production function and differentiate with respect to time to obtain (using the chain rule as shown in the previous section): (2.2) where is, is and is. Applying exactly the same technique as used above in deriving the expression for the growth rate of output, we obtain: (2.3) The growth of output is equal to a function of the growth rates of capital, labour and the technology factor. It is possible to simplify this if we make further assumptions about the market environment. This will enable it to be used to get an estimate of the final term, which is called total factor productivity growth, from data. We assume that labour and capital are traded in competitive markets and are paid their respective marginal products. This means that the marginal product of labour,, whereis the real wage, and that the marginal product of capital,

6 6 0. NOTE ON GROWTH AND GROWTH ACCOUNTING Total GDP Growth due to capital due to labour Solow residual TABLE 1. Solow Growth Accounting for the United States, Source: BLS, whereis the rental cost of capital in the economy. Additionally, we define the Solow residual by. We can now rewrite the expression above as: (2.4) Since corresponds to the share of total income spent by the economy on payments to capital it is often called the capital or profit share. Similarly, corresponds to the share of total income spent by the economy on payments to labour and is called the labour share. Hence, we have a compact expression for the Solow residual, which is also called total factor productivity growth as growth (Total factor productivity (TFP) growth: the Solow Residual) The Solow residual is the difference between output growth and a weighted sum of factor growths with weights given by the factor shares, i.e. it is the growth that is not attributed to the growth of either labour or capital inputs. If we accept the market assumptions we have used to derive the formula above we now have an expression that can be used to estimate the Solow residual (i.e. growth) from macroeconomic data. When the production function is constant returns to scale, the sum of the labour and capital shares is one and we can simplify equation (Total factor productivity (TFP) growth: the Solow Residual) to growth In empirical analysis, it is often illuminating to work in per capita terms and we can rearrange this equation as follows: growth where and. We can also turn the equation around and decompose the growth of productivity into the contribution from the growth of capital intensity and the contribution of TFP growth: growth. Growth accounting exercises may also seek to separately identify the contribution to the growth of output per worker from improvements in labour quality, e.g. as measured by increased average years of education. This will lead to a further component in the above equation: growth, where is the growth rate of the quality of labour inputs (per worker). This is weighted by labour s share in output, Example. In Table 1 we present an analysis of growth accounting as reported by the Bureau of Labor Statistics in the United States. It is interesting to note that according to growth accounting, the so-called New Economy phase of US growth in the late 1990s is associated with a higher contribution from factor input growth, especially capital, and faster TFP growth. TFP growth in the late 1990s is, however, not as fast as in the 1950s and 1960s.

### Handout on Growth Rates

Economics 504 Chris Georges Handout on Growth Rates Discrete Time Analysis: All macroeconomic data are recorded for discrete periods of time (e.g., quarters, years). Consequently, it is often useful to

### Introduction to the Economic Growth course

Economic Growth Lecture Note 1. 03.02.2011. Christian Groth Introduction to the Economic Growth course 1 Economic growth theory Economic growth theory is the study of what factors and mechanisms determine

### 6.2 The Economics of Ideas. 6.1 Introduction. Growth and Ideas. Ideas. The Romer model divides the world into objects and ideas: Chapter 6

The Romer model divides the world into objects and ideas: Chapter 6 and Ideas By Charles I. Jones Objects capital and labor from the Solow model Ideas items used in making objects Media Slides Created

### Cost-Volume-Profit Analysis

Cost-Volume-Profit Analysis Cost-volume-profit (CVP) analysis is used to determine how changes in costs and volume affect a company's operating income and net income. In performing this analysis, there

### 5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### 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

### Economics 304 Fall 2014

Economics 304 Fall 014 Country-Analysis Project Part 4: Economic Growth Analysis Introduction In this part of the project, you will analyze the economic growth performance of your country over the available

### 6. Methods 6.8. Methods related to outputs, Introduction

6. Methods 6.8. Methods related to outputs, Introduction In order to present the outcomes of statistical data collections to the users in a manner most users can easily understand, a variety of statistical

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1.

1 Growth rates Hats 1 are growth rates, or percentage changes, in any variable. Take for example Y, the GDP in year t compared the year before, t 1. We have: Ŷ = Y Y = Y t Y t 1 Y t 1 = Y t Y t 1 1 Example

### Notes 10: An Equation Based Model of the Macroeconomy

Notes 10: An Equation Based Model of the Macroeconomy In this note, I am going to provide a simple mathematical framework for 8 of the 9 major curves in our class (excluding only the labor supply curve).

### Cobb-Douglas Production Function

Cobb-Douglas Production Function Bao Hong, Tan November 20, 2008 1 Introduction In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an

### 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

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Financial Mathematics

Financial Mathematics 1 Introduction In this section we will examine a number of techniques that relate to the world of financial mathematics. Calculations that revolve around interest calculations and

### Preliminaries. Chapter Compound Interest

Chapter 1 Preliminaries This chapter introduces interest rates and growth rates. The two topics are closely related, so we treat them together. The concepts discussed here are not in Barro, but they will

### Examiner Tips for AS and A Level Physics (9702)

Examiner Tips for AS and A Level Physics (9702) How to Use These Tips These tips highlight some common mistakes made by students. They are collected under various subheadings to help you when you revise

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur. Lecture - 2 Simple Linear Regression

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 2 Simple Linear Regression Hi, this is my second lecture in module one and on simple

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### M248 Diagnostic Quiz

THE OPEN UNIVERSITY Faculty of Mathematics and Computing M248 Diagnostic Quiz Prepared by the Course Team [Press to begin] c 2005 The Open University Last Revision Date: March 17, 2006 Version 1.4 Section

### 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

### Macroeconomics Lecture 1: The Solow Growth Model

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

### Methods of Measuring Population Change. Measuring Population Change

Methods of Measuring Population Change CDC 103 Lecture 10 April 22, 2012 Measuring Population Change If past trends in population growth can be expressed in a mathematical model, there would be a solid

### Section 3-7. Marginal Analysis in Business and Economics. Marginal Cost, Revenue, and Profit. 202 Chapter 3 The Derivative

202 Chapter 3 The Derivative Section 3-7 Marginal Analysis in Business and Economics Marginal Cost, Revenue, and Profit Application Marginal Average Cost, Revenue, and Profit Marginal Cost, Revenue, and

### High School Algebra 1 Common Core Standards & Learning Targets

High School Algebra 1 Common Core Standards & Learning Targets Unit 1: Relationships between Quantities and Reasoning with Equations CCS Standards: Quantities N-Q.1. Use units as a way to understand problems

### The Macroeconomy in the Long Run The Classical Model

PP556 Macroeconomic Questions The Macroeconomy in the ong Run The Classical Model what determines the economy s total output/income how the prices of the factors of production are determined how total

### 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

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### 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

### Short-Run Production and Costs

Short-Run Production and Costs The purpose of this section is to discuss the underlying work of firms in the short-run the production of goods and services. Why is understanding production important to

### Reference: Gregory Mankiw s Principles of Macroeconomics, 2 nd edition, Chapters 10 and 11. Gross Domestic Product

Macroeconomics Topic 1: Define and calculate GDP. Understand the difference between real and nominal variables (e.g., GDP, wages, interest rates) and know how to construct a price index. Reference: Gregory

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

### 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

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### Employment, unemployment and real economic growth. Ivan Kitov Institute for the Dynamics of the Geopsheres, Russian Academy of Sciences

Employment, unemployment and real economic growth Ivan Kitov Institute for the Dynamics of the Geopsheres, Russian Academy of Sciences Oleg Kitov Department of Economics, University of Oxford Abstract

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which

### Before Growth: The Malthusian Model

University College Dublin, Advanced Macroeconomics Notes, 2015 (Karl Whelan) Page 1 Before Growth: The Malthusian Model We have devoted the last few weeks to studying economies that grow steadily over

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### ( ) = ( ) = ( + ) which means that both capital and output grow permanently at a constant rate

1 Endogenous Growth We present two models that are very popular in the, so-called, new growth theory literature. They represent economies where, notwithstanding the absence of exogenous technical progress,

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Graphical Integration Exercises Part Four: Reverse Graphical Integration

D-4603 1 Graphical Integration Exercises Part Four: Reverse Graphical Integration Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Laughton Stanley

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that

### 01 In any business, or, indeed, in life in general, hindsight is a beautiful thing. If only we could look into a

01 technical cost-volumeprofit relevant to acca qualification paper F5 In any business, or, indeed, in life in general, hindsight is a beautiful thing. If only we could look into a crystal ball and find

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Chapter 12: Cost Curves

Chapter 12: Cost Curves 12.1: Introduction In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level

### 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

### 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

### (Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

### 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* =

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

### 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

### Math 016. Materials With Exercises

Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of

### Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.

Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections Least-Squares Regression Linear Regression General Linear Least-Squares Nonlinear

### Items related to expected use of graphing technology appear in bold italics.

- 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### Solution to Individual homework 2 Revised: November 22, 2011

Macroeconomic Policy Fabrizio Perri November 24 at the start of class Solution to Individual homework 2 Revised: November 22, 2011 1. Fiscal Policy and Growth (50p) After reviewing the latest figures of

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### 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)

### PЄ d = Percentage change in Quantity Demanded Percentage change in Price

Lesson 0 ELASTICITIES IMPORTANCE OF ELASTICITY IN OUR TODAY S LIFE There is much more importance of the concept of elasticity in our life. The firm which uses advertising to change prices uses the concept

### The Market-Clearing Model

Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

### 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

### Rules for Exponents and the Reasons for Them

Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication

### Notes 5: More on regression and residuals ECO 231W - Undergraduate Econometrics

Notes 5: More on regression and residuals ECO 231W - Undergraduate Econometrics Prof. Carolina Caetano 1 Regression Method Let s review the method to calculate the regression line: 1. Find the point of

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### University of Lethbridge Department of Economics ECON 1012 Introduction to Microeconomics Instructor: Michael G. Lanyi. Chapter 22 Economic Growth

University of Lethbridge Department of Economics ECON 1012 Introduction to Microeconomics Instructor: Michael G. Lanyi Chapter 22 Economic Growth 1) Economic growth is A) equal to real GDP per capita multiplied

### CHAPTER 2 Estimating Probabilities

CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a

### Continuous Functions, Smooth Functions and the Derivative

UCSC AMS/ECON 11A Supplemental Notes # 4 Continuous Functions, Smooth Functions and the Derivative c 2004 Yonatan Katznelson 1. Continuous functions One of the things that economists like to do with mathematical

### 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

C H A P T E R 3 Simulation of Business Processes The stock and ow diagram which have been reviewed in the preceding two chapters show more about the process structure than the causal loop diagrams studied

### CHAPTER 3. Exponential and Logarithmic Functions

CHAPTER 3 Exponential and Logarithmic Functions Section 3.1 (e-book 5.1-part 1) Exponential Functions and Their Graphs Definition 1: Let. An exponential function with base a is a function defined as Example

### 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

### Nominal, Real and PPP GDP

Nominal, Real and PPP GDP It is crucial in economics to distinguish nominal and real values. This is also the case for GDP. While nominal GDP is easier to understand, real GDP is more important and used

### 7: Compounding Frequency

7.1 Compounding Frequency Nominal and Effective Interest 1 7: Compounding Frequency The factors developed in the preceding chapters all use the interest rate per compounding period as a parameter. This

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Exponentials. 1. Motivating Example

1 Exponentials 1. Motivating Example Suppose the population of monkeys on an island increases by 6% annually. Beginning with 455 animals in 2008, estimate the population for 2011 and for 2025. Each year

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### Number of Workers Number of Chairs 1 10 2 18 3 24 4 28 5 30 6 28 7 25

Intermediate Microeconomics Economics 435/735 Fall 0 Answers for Practice Problem Set, Chapters 6-8 Chapter 6. Suppose a chair manufacturer is producing in the short run (with its existing plant and euipment).

### 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

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### Regression Analysis. Data Calculations Output

Regression Analysis In an attempt to find answers to questions such as those posed above, empirical labour economists use a useful tool called regression analysis. Regression analysis is essentially a

### So we can use either equilibrium condition to get the same result.

Numerical Problems 1. (a) S d = Y C d G = Y (4000 4000r + 0.2Y) 2000 = 6000 + 4000r + 0.8Y. (b) (1) Using the equation that goods supplied equals goods demanded gives Y = C d + I d + G = (4000 4000r +

### High School Algebra Reasoning with Equations and Inequalities Solve systems of equations.

Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student

### Fourier Series, Integrals, and Transforms

Chap. Sec.. Fourier Series, Integrals, and Transforms Fourier Series Content: Fourier series (5) and their coefficients (6) Calculation of Fourier coefficients by integration (Example ) Reason hy (6) gives

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### 2.4 Motion and Integrals

2 KINEMATICS 2.4 Motion and Integrals Name: 2.4 Motion and Integrals In the previous activity, you have seen that you can find instantaneous velocity by taking the time derivative of the position, and

### Chapter 13. The annuity mortgage Interest discrete versus continuous compounding. Reading material for this chapter: none recommended

Chapter 13 The annuity mortgage Reading material for this chapter: none recommended We formulate a mathematical model for the simplest possible mortgage, the annuity mortgage. We assume that interest is

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Repton Manor Primary School. Maths Targets

Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work

### Chapter 14: Production Possibility Frontiers

Chapter 14: Production Possibility Frontiers 14.1: Introduction In chapter 8 we considered the allocation of a given amount of goods in society. We saw that the final allocation depends upon the initial

### Chapter 1: The binomial asset pricing model

Chapter 1: The binomial asset pricing model Simone Calogero April 17, 2015 Contents 1 The binomial model 1 2 1+1 dimensional stock markets 4 3 Arbitrage portfolio 8 4 Implementation of the binomial model

### Review Notes The Demand for Labor Elasticities

Review Notes The Demand for Labor Elasticities Wage elasticity of the demand for labor = η w What is the impact of the minimum wage on labor markets? Recall that if the market is competitive then the wage

### The Economy as a Whole

V The Economy as a Whole 12 Models of the whole economy 12.1 INTRODUCTION Running a model of the UK economy can be a large task, requiring considerable financial support. There are thus a fairly small