ch03-exponential-logarithmic-functions

ch03-exponential-logarithmic-functions - Exponential and...

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Unformatted text preview: Exponential and Logarithmic Functions 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 X t after 1 time period: X t+1 = (1 + r)X t – r = 10%; \$10 today is worth (1.1)10 = \$11 next year • Value of X t after 2 time periods: X t+2 = (1 + r)X t+1 = (1 + r)(1 + r)X t = (1 + r) 2 X t • Value of X t after n time periods X t+n = (1 + r)X t+n-1 = (1 + r) n X t • \$1 earning 5% for 50 years = \$11.47 • \$1 earning 10% for 50 years = \$117.39 – Doubling interest rate has a huge impact Exponential growth 10 20 30 40 50 r=5% r=10% n (years) (1 ) n r + More frequent compounding • Once per year: • Twice per year: • k times per year: 2200 ∞ times per year: (1 ) r + 2 2 (1 ) r + 1 1 (1 ) (1 ) (1 ) k r k r r k mr r k m + = + = + ( 29 1 lim(1 ) r m r m m e →∞ + = 1 2 10 2.593742 100 2.704814 10000 2.718146 100000000 2.718282 1 (1 ) k k + k Most important constant in economics 2.718281828459045235360287471352662497757247093699 959574966967627724076630353547594571382178525166 427427466391932003059921817413596629043572900334 295260595630738132328627943490763233829880753195 251019011573834187930702154089149934884167509244 761460668082264800168477411853742345442437107539 077744992069551702761838606261331384583000752044...
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ch03-exponential-logarithmic-functions - Exponential and...

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