Econ 300_Summer 2009_Slides 2-exponential-logarithmic[1]

# Econ 300_Summer 2009_Slides 2-exponential-logarithmic[1] -...

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Unformatted text preview: 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 alue of X fter n time periods • 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 5% for 50 years = \$11.47 • \$1 earning 10% for 50 years = \$117.39 oubling interest rate has a huge impact – Doubling interest rate has a huge impact Exponential growth 50 (1 ) n r g14 30 40 r=10% 20 10 r=5% 10 20 30 40 50 n (years) More frequent compounding • Once per year: (1 ) r g14 ) • Twice per year: • k times per year: 2 2 (1 ) r g14 1 1 (1 ) (1 ) (1 ) k r k r r k mr r k m g14 g32 g14 g32 g14 • g102 times per year: g11 g12 1 lim(1 ) r m r m m e g111g102 g14 g32 1 (1 ) k k g14 k 1 2 10 2.593742 100 2.704814 0000 718146 10000 2.718146 100000000 2.718282 Most important constant in economics 1 lim(1 ) k k k e g111g102 g32 g14 g32 2.718281828459045235360287471352662497757247093699 959574966967627724076630353547594571382178525166 427427466391932003059921817413596629043572900334 295260595630738132328627943490763233829880753195 51019011573834187930702154089149934884167509244 251019011573834187930702154089149934884167509244 761460668082264800168477411853742345442437107539 077744992069551702761838606261331384583000752044 33826560297606737113200709328709127443747047230 933826560297606737113200709328709127443747047230 696977209310141692836819025515108657463772111252 389784425056953696770785449969967946864454905987 9316368892300987931 Exponential growth...
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Econ 300_Summer 2009_Slides 2-exponential-logarithmic[1] -...

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