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This 2 page Class Notes was uploaded by Miss Sigurd Dicki on Wednesday September 23, 2015. The Class Notes belongs to ECON202 at Drexel University taught by ChristopherLaincz in Fall. Since its upload, it has received 20 views. For similar materials see /class/212540/econ202-drexel-university in Economcs at Drexel University.
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Date Created: 09/23/15
Calculating Growth Rates There are two ways to calculate growth rates The rst uses the average growth per time period The second uses continuous growth Method 1 The average growth rates of economic variables can be computed using the following formula K Y01 x 1 where Y is the value of Y at time t Yb is the initial value of Y and at is the average annual growth rate For example suppose you are asked to calculate the average annual real GDP per capita what does per capita mean growth rate of the United States between 1950 and 2000 Real GDP in 1950 was 11921 and in 2000 it was 32629 Thus we have Y0 11921 Y 32629 and t 50 Putting these values into the above equation we have 32629 119211 9 2 at is the average annual growth rate Divide both sides by 11921 raise both sides to the power 150 and then subtract 1 to get 32 629 x 11 921 To solve for at we could also use natural logs ln There are many useful properties of natural logs which apply here 150 7 1 00203 or 203 3 lnab lna lnb and ln lna 7111b 4 lnab blna 5 51quot a 6 Taking the natural logs of both sides of 2 we have ln32 629 ln 119211 x5 Applying the property of natural logs in 4 above we have ln32629 ln 11 921 ln1 9 Applying the property 5 we get ln32629 ln 11921 501n1 Algebraic rearranging gives us ln1 ac ln32629 7 ln 11 921 Using another property of natural logs 4 we can write ln1 at 51 0 1n Now you do need a calculator here to nd that ln1 ac 10069 00201 Using the last property of natural logs 6 by putting both sides of the above into an exponent over base 5 we get eln1w 60 02017 A 1 x 60 0201 Using your calculator again 1 as 10203 or at 00203 which means the average annual growth rate was 203 Method 2 For the second method the standard formula is Y Yoem where Y is the value of Y at time t Yb is the initial value of Y and g is the growth rate The difference between this version and Method 1 is that growth is continuously compounded For example if initial GDP is 1000 the growth rate is 2 per year then the value of Y after 30 years will be Yt 1000e 02 182212 Suppose you want to know how long it will take to double GDP when the annual growth rate is 2 Then you can solve for t in the following manner using lns Y Yoem Take lns of both sides to get ln ln Y0 ln e91 which is the same as lnY lnY0 gtlne Since lne 1 we have 1nltm1nltYo gt Using algebra to nd t we get 1 t lnY 7 lan 9 Plugging in the values we nd t t ln 2000 7 ln 1 000 3466 years 7 1 7 002 One more example What growth rate would need to be achieved for Y to double in 25 years Again starting from K Yoeg we need to solve for g in this case Taking lns we have ln lnY0 gt which yields 9 lnY ilan Plugging in the values 9 ln2000 71n1000 00278 or 278 1 g The two methods will give slightly different answers The reason is as follows Consider a bank account that you have which currently holds 1000 and pays an interest rate of 2 per year How the value of that account grows depends on whether the interest is paid at the end of the year or if the interest is added continuously at all points in time If it is only added at the end of the year you will get 20 after one year for a total of 1020 in your bank account On the other hand if it is added continuously you will have earned some interest after holding it in the bank every second of every day although this amount will be very small However after say one month interest will be paid on your original 1000 plus you will be paid interest on the interest you earned in the rst month That implies that you earn more overall with quot A 39 Under 39 A quot you will have 102020 after one year A small difference over a shorteperiod but over a long period the difference grows
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