San Francisco State University Michael Bar ECON 60 Summer 2018 Due Wednesday, July 11 Problem set 1 Name Assignment Rules 1. Homework assignments must be typed. For instructions on how to type equations and math objects please see the notes Typing Math in MS Word. 2. Homework assignments must be prepared within this template. Save this file on your computer and type your answers following each section. 3. Do not delete the questions. 4. Late homework assignments will not be accepted under any circumstances. All the graphs should be fully labeled, i.e. with a title, labeled axis and labeled curves. 6. For instructions on how to plot data with Excel, see the notes Instructions for Excel. 7. In all the questions that involve calculations, you are required to show all your work. That is, you need to write the steps that you made in order to get to the solution. 8. This page must be part of the submitted homework. 1
The Mathematics of Growth Rates 1. Suppose that you deposit a certain amount of money in a savings account with interest rate of 2% per year. a. How long will it take your money to double? Show your calculations. You are not allowed to use any approximation formulas, such as the rule of 70. Round your answer to decimal places. Let yy 0 be the initial amount of your deposit. Doubling this amount means that after certain time you have 2yy 0. Thus, we need to solve the following equation for the unknown t: yy 0 (1.02) tt = 2yy 0 (1.02) tt = 2 Observe that yy 0 cancels out, which means that the doubling time is independent of the initial value. Solving the above: tt ln(1.02) = ln(2) tt = ln(2) = 3.00279 yyyyyyyyyy ln(1.02) b. Calculate the approximate doubling time using the rule of 70. tt 70 2 = 3 yyyyyyyyyy 2. Suppose that U.S. GDP is twice as large as the GDP of China. Suppose that the U.S. GDP grows on average at 2% per year, while the Chinese GDP grows at 7% per year. How many years would it take China to catch up with the U.S. in terms of GDP? Let xx be the GDP in China, so that 2xx is the U.S. GDP. If after tt years China catches up, we need to solve the following equation for the unknown tt: xx (1.07) tt = 2xx (1.02) tt Notice that xx cancels out, which means that the catchup time is independent of the actual levels, but only of the ratio and growth rates. Solving the above: tt ln(1.07) = ln(2) + tt ln(1.02) tt[ln(1.07) ln(1.02)] = ln(2) ln(2) tt = = 14.484 yyyyyyyyyy ln(1.07) ln(1.02) Remark: approximately, the above is equivalent to 70 = 14 yyyyyyyyyy, as China catches up at % relative to U.S. and needs to double relative to U.S. 3. Suppose that the price of the firm s output has increased by 1.%, and the quantity sold increased by 2.%. What is the approximate growth rate of the firm s revenue? RR = PP QQ RR PP + QQ = 1.% + 2.% = 4% 2
4. Suppose that the GDP of some country grows at 3% per year, and its population grows at 2.% per year. What is the approximate growth rate of GDP per capita in this country? GGGGGG PPPPPP GGGGGG PPPPPP = 3% 2.% = 0.%. Suppose that some variable is growing at constant rate. a. Prove that the natural logarithm of that variable is a linear function of time. If a variable yy grows at constant rate gg, then its value at time tt is yy tt = yy 0 (1 + gg) tt Taking logs: ln(yy tt ) = ln(yy 0 ) + tt ln(1 + gg) Thus, ln(yy tt ) is a linear function of time, with intercept ln(yy 0 ) and slope ln(1 + gg). b. Find the intercept and slope of the linear function in part a. The intercept is ln(yy 0 ) and the slope is ln(1 + gg). 6. Chose the correct answer and provide a mathematical proof. a. ln(80) ln(40) > ln(10) ln () b. ln(80) ln(40) < ln(10) ln () c. ln(80) ln(40) = ln(10) ln () ln(80) ln(40) = ln 80 = ln(2) 40 ln(10) ln(0) = ln 10 = ln(2) Thus, the difference of ln() of two variables depends on the ratio of the two variables. For example, each time we double some variable, the ln of that variable increases by a constant number ln(2) 0.7, and if the variable triples, the ln of the variable increases by a constant ln(3) 1.1, etc. 7. This question illustrates that when a variable grows at constant rate, then the graph of the ln of the variable is a linear function of time, with slope that is approximately equal to the growth rate of the original variable (when that growth rate is small). Suppose that you put 100$ in a savings account at annual interest rate of 3.%. Let SS tt be the amount of savings at time t, where tt = 0,1,,20. a. Using Excel, plot the graph that shows the amount of savings that you have in each of the years tt = 0,1,,20. That is, plot SS tt against tt. 3
200 Savings over time 190 180 170 160 S_t 10 140 130 120 110 100 0 2 4 6 8 10 12 14 16 18 20 t b. Using Excel, plot the graph of ln (SS tt ) against tt. ln(savings).4.3.2.1 ln(s_t) 4.9 4.8 4.7 4.6 4. 0 10 1 20 2 t c. Describe the shape of the graph in part b. What is the exact slope of the graph in part b? The ln (SS tt ) is a linear function of time. The exact slope of the line is ln(1.03) = 0.0344, which is approximately the (net) growth rate of the original variable SS tt. 4
8. Based on the following graph, what is the approximate growth rate of the variable y t? Circle the correct answer, and briefly explain. 2.6 2.6 2. 2. ln(y_t) 2.4 2.4 2.3 2.3 2.2 0 2 4 6 8 10 12 Time a. 2.3% b. 2.6% c. 10% d. 3% To see why, recall that the slope of the ln(yy tt ) is approximately equal to the growth rate of yy tt. The slope of the above line is ΔYY 2.6 2.3 = = 0.3 = 0.03 = 3% ΔXX 10 10 Introduction 9. Based on the graphs from the Introduction slides that displays the real GDP per capita and the ln of that variable in different countries (ARG = Argentina, CHN = China, KOR = South Korea, TCD = Chad, USA = USA). a. Are all the above countries experiencing sustained growth in real GDP per capita? 1 No. Argentina had negative growth in real GDP per capita during the 80 s. Chad experienced negative growth most of the time. 1 Sustained growth means positive and constant growth rate throughout the entire period.
b. What can you say about the growth rate of real GDP per capita in the USA and Argentina prior to 197? Explain how you reached your conclusion. About the same. By looking at the ln(real GDP per capita) of both countries, we observe that before the mid-seventies, the two trends were approximately parallel and straight lines. This means that the slopes of the trends were the same, and thus the growth rate was about the same. 10. This is a small research project. Visit the CIA World Factbook, ranking of countries by their real GDP per capita. a. Choose countries among the 30 poorest and countries among the 30 richest, based on GDP per capita. For each country report (1) RGDP per capita, (2) TFR (Total Fertility Rate, which is the total number of children that a woman is expected to have during her lifetime), (3) life expectancy, and (4) one more variable that you think is interesting. Present your results in two tables, one for the rich countries and one for the poor countries, with the averages for each group of countries. In other words, fill in the information in the tables below, with your choice of countries. Poor Countries: RGDP/capita TFR Life Female Expectancy Literacy Rate Zimbabwe $1,700 3.0 8.00 84.6% Afghanistan $2,000.22 1.30 24.2% Madagascar $1,00 4.12 6.90 62.6% Haiti $1,800 2.79 63.80 7.3% Somalia $400.89 2.40 2.8% Average $1480 4.304 8.28 0.90% Rich Countries: RGDP/capita TFR Life Female Expectancy Literacy Rate Norway $69,300 1.86 81.80 100% U.S. $7,300 1.87 79.80 99% Hong Kong $8,100 1.19 82.90 99% Australia $48,800 1.77 82.20 99% Qatar $129,700 1.90 78.70 96.8% Average $72,640 1.718 81.08 98.76% b. What did you learn from this mini-research about the correlation between standard of living (real GDP per capita), and other variables in the table? I learnt that poor countries/regions tend to have higher fertility, lower life expectancy and lower female literacy rates. I was surprised by the high female literacy rate in Zimbabwe one of the poorest countries in the world today. 6