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# Calculus I MATH 115

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This 5 page Class Notes was uploaded by Maggie Casper on Thursday October 29, 2015. The Class Notes belongs to MATH 115 at Wellesley College taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/230918/math-115-wellesley-college in Mathematics (M) at Wellesley College.

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Date Created: 10/29/15

Fall Semester 05 06 Akila Weempana Lecture 2 Mathematical Preliminaries 1 OVERVIEW 0 This course will require you to use basic calculus skills The more comfortable you are with the material covered in Math 115 the easier you ll nd this course to be If it has been a while since you took calculus I urge you to review some basic concepts Since the topic at hand deals with economic growth the basic mathematical foundations we need relate to calculating growth ratesln particular we will learn how to do four basic tasks related to economic growth 0 First given a a period of time b the average growth rate over that period of time and c the initial or terminal value of a variable we will learn to calculate the terminal or initial value of that variable In other words given a initial value and a growth rate we can gure out what the value will be at some point in the future or equivalently given the ending value and a growth rate we can gure out what the initial value must have been For this task we make use of a special function in mathematics known as the exponential function 0 Second we will do a slight variation of that problem Given a a period of time b an initial value and c a terminal value for a variable we will learn to calculate the average growth rate of that variable over the period In other words given a initial value and an ending value we can gure out how fast the variable grew on average over the period For this we make use of another special function in macro known as the logarithmic function Next we will show how given an algebraic equation that describes how the level of a variable changes with time we will derive an algebraic equation for the growth rate of that variable at a given point in time 0 Finally we will show how given a relationship among certain variables say Z fX Y we can calculate the growth rate of a variable Z as a function of the growth rates of X and Y II CALCULATING THE FUTURE OR PRESENT VALUE Discrete Compounding 0 Under discrete compounding a variable x changes in value at distinct points during a period c We rst consider the simplest case in which the variable changes only once at the end of the period 0 Suppose you are given the initial value mo and the per period growth rate r of a variable m The value of z after 1 period will be 1 z01 r and the value of z after 2 periods will be 2 z11 r z01 r2 Extending this we can show that after t periods the future or compounded value of x can be expressed as mt z01 Tt 0 Similarly if we knew the value of x at time t we can calculate the initial or discounted value of that variable as 0 0 Now let s make things a little more complicated by supposing that x was still growing at the rate of r a year but the compounding was happening k times a year Ex a bank account that pays interest monthly at an interest rate of 5 a year has r005 and k12 0 Now the value of z after 1 period will be 1 z01 Generalizing we can show that after t periods the future or compounded value of x can be expressed as mt mo 1 a 0 Similarly if we knew the value of x at time t we can calculate the initial or discounted value of that variable as 0 m M 1 The Exponential Function 0 An exponential function is a function of the form y aa m where a 04 and B are constants ln economics what we call the exponential function is one in which a e ie functions of the form y 045 where the number e is a very special number in mathematics de ned as limH00 1 i approximately equaling 2718 Some useful rules of the exponential function to brush up on are the following 5am gtlt elm eabm 5am elm 6a7bm eazn enam o The exponential function is vital in calculating the initial or terminal value of a variable whose value is changing continuously over time Continuous Compounding 0 Under continuous compounding a variable x is changing in value all the timeln other words unlike your bank account on which interest may be calculated annually or quarterly or monthly or daily many economic variables are changing at every instant in time c We can think of continuous compounding as a special case of discrete compounding where k approaches in nity So the value t periods from now of a variable that is being compounded continuously at a rate r can be expressed as 3 132on 1 f c We can do a little bit of algebra and re write this as E 1 T mt lim m0 1 we mt m0 5 39rt 0 This simpli es to 0 Suppose you are given the initial value mo and the per period growth rate r of a continuously growing variable x The value of z after t periods is mt 05 0 Similarly if we knew the value of x at time t we can calculate the initial or discounted value of that variable as 0 we Example 0 Suppose you put 15000 in a bank account that pays a 5 annual rate of return Calculate the value of the bank balance after 1 3 and 5 years under discrete and continuous compounding 0 Under discrete compounding Bt B01 T S0 B1 15750 B3 1736438 and B5 1914422 0 Under continuous compounding Bt B05quot So B1 1576907 B3 17 42751 and B5 19 26038 III CALCULATING THE GROWTH RATE OF A VARIABLE The Logarithmic Function 0 The logarithmic function in mathematics is the inverse of an exponential function So the inverse of the exponential function y b1 is the logarithmic function x logby also called the value of y in log base b o In economics what we call the logarithmic function is typically the natural logarithmic function ie logarithms in base e So the inverse of the function y e1 is the logarithmic function x lny Some useful rules of the logarithmic function to remember are the following lnxy 1m lny ln 1m 71mg 111w nlnm ln EM 35 51nfw 35 Using the Logarithm to Calculate Growth Rates 0 The growth rate of a variable refers to the percentage rate of change of that variable over a given period of time In most economic applications especially in macroeconomics growth rates are expressed in terms of the percentage rate of change over a year So if the period of time is longer than a year we typically express the average growth rate ie by what the variable has changed every year on average over the period The average growth rate of GDP from 1960 1970 is the increase in GDP per year on average from 1960 70 Given observations of a variable X at two points in time let s say year 0 and year t we can calculate the average annual growth rate of the variable as lnXt 7 lnX0 t IV In other words the growth rate can be found by taking the difference in the natural logs of the variable divided by the elapsed number of years Why does this formula work Think about the following example Suppose the capital stock grows at a rate g per period and that we observe the value of the capital stock at two periods in time n periods apart which we denote by K0 and Kt The relationship between K0 and Kt can be expressed as follows under discrete compounding Kt K01 gt and under continuous compounding Kt K0591 Taking logs of the expression we got under continuous compounding we get lnKt lnK0 lnegt E lnK0 gt This simpli es down to lnKt 7 lnK0 gt or equivalently that w g The difference in natural logs divided by the elapsed number of periods gives us the exact average per period growth rate of the capital stock Now consider the case of discrete compounding Since Kt K01 gt lnKt lnK0 tln1 g This simpli es down to w ln1 g This time taking the difference in natural logs and dividing by the elapsed number of periods does not give us the exact per period average growth rate of the capital stock However as long as g is not too large the value of ln1g is very close to g Try it out and see for yourself So the difference in natural logs divided by the number of periods elapsed gives an exact answer for the average per period growth rate for continuously compounded variables and an approximate answer for the average per period growth rate for discretely compounded variables Note that describing the average rate of growth over a period of many years in time does not mean that the variable always grew at the same rate It may have grown faster than the average in some periods slower than the average in others Think back to the graph of Us Real GDP I showed in class on the rst day on average the Us economy grew at a rate of 325 a year but the graph clearly shows that in some years growth was higher and in other years growth was lower but the average still has a lot of information for us TIME DERIVATIVES AND GROWTH RATES In the models that we develop in this class we will often have to derive an expression for the growth rate of a continuously growing variable at some particular instant in time Consider an economic variable that changes continuously over time eg the capital stock K Use the following short hand notation to denote the derivative of K with respect to time then represents the instantaneous change in the variable ie by how much K changes in the next instant of time The expression K represents by what K changes in K the next instant of time or equivalently the growth rate of K In other words given an algebraic expression for how a variable evolves with time we can calculate an algebraic expression for the instantaneous growth rate of that variable by taking the derivative of K with respect to time and dividing by K 0 Another very useful result is that dlnK 7 dlnKdK 7 id 7 5 dt dK E KW K Basically7 given an algebraic expression for a variable K7 we can calculate an algebraic ex pression for the growth rate of that variable by taking the derivative of lnK with respect to time So if you take the derivative of the log of K with respect to time you no longer need to divide by K to get an expression for the growth rate 0 In this class7 we will typically express growth rates on a per year basis Therefore7 the expression 005 means that the the growth rate of the variable K at the current instant in time is a rate of 5 per year THE TLAD TECHNIQUE o Occasionally7 we come across economic variables that are related to one another in certain ways If we know how these variables are related7 we can nd a relationship that should hold among the growth rates of the variables using a technique that we will term the 77Take Logs and Differentiate77 TLAD technique 0 This technique will prove enormously helpful in nding relationships between the growth rates of various variables We can show that the following rules hold 1 lf 2 a and g b then the growth rate of C AB is a b This can be seen by the following algebra g 7 dlnC 7 dlnAB 7 dlnA 7 dlnB 0 dt dt dt dt 7 ABi 12 A B 2 If g a and g b then the growth rate of D g is g a 7 b This can be seen by the following algebra Q dlnD 7 dln g 7 dlnA dlnB D dt dt dt dt 3 If g a and 04 is an arbitrary constant then the growth rate of F A is aa This can be seen by the following algebra 5 7 dlnF7dlnAD adlnA F at dt dt

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