Properties of Growth Rates • Growth rates: how they work 1 o This allows us to calculate average growth rates over long period of time. /1 Rule of 70 21 21 2 1 2 1 1 0.7 70 100 • Suppose a quantity is growing at a constant rate. To figure out the number of years for it to double, divide 70 by the growth rate times 100. • So if it’s growing at 2% per year, it will double every ________ years, regardless of the starting point (the original xt value cancelled out). Ratio (logarithmic) scales Ratio scales allow one to see data more easily if their rate of growth is constant. These graphs represent 2n, where n is the number on the horizontal axis. 01000020000300004000050000600007000002468101214161101001000100001000000246810121416Year US Income Average growth rate 1870 2525 1929 7,100 71002525/1 =1.77% 1950 11,720 117207100/1 = 2004 36,880 _/ 1 =2.15% ln ln ln ln Property of logarithms Exercise Start with a bank account balance of $100. At times 0, 1, 2, 12, 24, 48, 60, what is the bank balance? Time Interest rate = 1% Interest rate = 10% 0 100*(1+0.01)0= 1 100*(1+0.10)1= 2 12 24 48 60 Plot the bank balances both on a standard scale and on a ratio scale Rules of Growth Rates 1. If , then 2. If , then 3. If , then Suppose 5% and 2% Calculate the growth rate of z if a. z=xy, b. z=x/y, c. z=x1/3y2/3 01002003004005000123456789101112131415161001000012345678910111213141516A CobbDouglas Production Function in Growth Rates Suppose that grows at a rate of grows at grows at Consider this production function Then the growth rate of Y is given by the growth rate of the right hand side of the equation. Using the second rule Using the third rule So output‐per‐worker grows because productivity gr ows or because capital‐per‐worker (which is subject to diminishing returns, since 1) grows. Consider this production function Then Using the second rule Using the third rule 1 So total output grows because productivity grows or because capital (which is subject to diminishing returns, since 1) grows or because population (also subject to diminishing returns) grows. Rearranging 1 “Growth Rate” Version of the Cobb‐Douglas Production Function
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