Exponential and Logarithmic Functions: Understanding Growth and Compounding - Prof. Erkut , Study notes of Economics

Download Exponential and Logarithmic Functions: Understanding Growth and Compounding - Prof. Erkut and more Study notes Economics in PDF only on Docsity! 1 Exponential and Logarithmic Functions (Klein chapter 3) Professor Erkut Ozbay Economics 300 Modeling growth • Exponential functions – Constant percentage growth per unit time • Logarithmic functions Growth of money • Interest rate r • Value of Xt after 1 time period: Xt+1 = (1 + r)Xt – r = 10%; $10 today is worth (1.1)10 = $11 next year • Value of Xt after 2 time periods: Xt+2 = (1 + r)Xt+1 = (1 + r)(1 + r)Xt = (1 + r)2Xt • Value of Xt after n time periods Xt+n = (1 + r)Xt+n-1 = (1 + r)nXt • $1 earning 5% for 50 years = $11.47 • $1 earning 10% for 50 years = $117.39 – Doubling interest rate has a huge impact 2 Exponential growth 10 20 30 40 50 10 20 30 40 50 r=5% r=10% n (years) (1 )nr+ More frequent compounding • Once per year: • Twice per year: • k times per year: • ∞ times per year: (1 )r+ 2 2(1 ) r+ 1 1(1 ) (1 ) (1 ) k r k r rk mrr k m+ = + = + ( )1lim(1 ) rm rmm e→∞ + = 1 2 10 2.593742 100 2.704814 10000 2.718146 100000000 2.718282 1(1 )kk+k Most important constant in economics 2.718281828459045235360287471352662497757247093699 959574966967627724076630353547594571382178525166 427427466391932003059921817413596629043572900334 295260595630738132328627943490763233829880753195 251019011573834187930702154089149934884167509244 761460668082264800168477411853742345442437107539 077744992069551702761838606261331384583000752044 933826560297606737113200709328709127443747047230 696977209310141692836819025515108657463772111252 389784425056953696770785449969967946864454905987 9316368892300987931 1lim(1 )kkke →∞= + = 5 Example • Perpetuity: value of $1 each period forever • r = 10%; δ = .909; perpetuity = $11.00 Discount rate 0 1 Discount factor 1 1 r r δ > = < + 2 2 1 ... ... , subtracting yields (1 ) 1 1 1 d d d d δ δ δ δ δ δ δ ∞ ∞ ∞ ∞ = + + + = + + − = = − Example • Annuity: value of $1 each period for n periods • r = 10%; δ = .909; 20-year annuity = $9.36 Discount rate 0 1 Discount factor 1 1 r r δ > = < + 2 1 2 1 ... ... , subtracting yields (1 ) 1 1 1 n n n n n n n n d d d d δ δ δ δ δ δ δ δ δ δ δ −= + + + + = + + + − = − −= − Logarithms • Inverse of exponential function • b is the base • Most commonly b = 10 or b = e • log base e is called natural logarithm: log ( ) finds exponent y such that yby x b x= = ln( )y x= 6 Log is inverse of exponential function Log base 10 • Example of log base 10 • x 1 10 100 1000 10000 • y 0 1 2 3 4 • Examples of log scales – Shock waves (Richter scale for earthquakes) – Sound waves (decibels for sound) – Radio waves (Hz, kHz, MHz, GHz) 10 10 log ( ) yx y x = = Log base 10 and base 2 7 1 2 3 4 5 10 100 1000 104 105 Log plot of exponential growth 10xy = xy e= 2xy = Properties of logarithmic functions log 1 log log log log log log log log log log log b b b b b b b y b b a b a b xy x y x x y y x y x x x b = = + = − = = Natural logs (base e) • Continuous growth models • Same properties hold • Example: Yahoo Finance (plotting stock history) ln 1 ln ln ln ln ln ln ln lny e xy x y x x y y x y x = = + = − = 10 Cobb-Douglas production • We measure Q, L, and K at each time: • Taking logs: • Nice linear model! • Can estimate parameters with econometrics ln ln ln ln t t t t t t t t Q A L K Q A L K α β α β = = + + How long does it take for something to double? • With r = 10% it takes 7 years for value to double • With r = 5% it takes 14 years for value to double • Moore’s Law of electronics: a doubling every 18 months – r = .6931/1.5 = 46% 0 0/ 2 ln( ) ln(2) ln(2) .6931 rn n rn n rn V e V e V V e n r r = = = = = = 11 Properties of logarithmic functions log 1 log log log log log log log log log log log b b b b b b b y b b a b a b xy x y x x y y x y x x x b = = + = − = = Exercise 3.3 1 Simplify 10log (100) ln 10 5 2 5 10 ln 1log 1ln x xe e x x y e α β− −       100= 0x x= − = 5 10 10log 5logx x −= = − 5 2( ln ln )x yα β= − + − Exercise 3.3 11 • $10,000 invested at 5% with continuous compounding • When do you have $15,000? • Use formula for future value: Xt+n = Xt ern • 15,000 = 10,000 e .05n • Solve for n e.05n = 15,000/10,000 = 1.5 .05 n = ln(1.5) ⇒ n = ln(1.5)/.05 = 8.1093