Econ 299 Chapter2a - 2 Mathematical Versions of Simple...

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1 2. Mathematical Versions of Simple Growth Models Simple growth models Examples of growth models Basic theory of econ models Calculus Review
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2 2. Mathematical Versions of Simple Growth Models 2.1 An Introduction to mathematical models of growth 2.2 Mathematical models of economic relationships 2.3 Interpreting parameters 2.4 Error terms: adding uncertainty to an economic model 2A Economic Applications of Calculus
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3 2.1.1 – Simple Models of growth Constant growth rate g Economic reports generally deal with growth rates (negative or positive) -Growth rates carry more intuitive sense than raw data From before: 1) Percentage growth = 100[X t -X t-1 ]/X t-1 =100g 2) % growth = 100 [ln(X t )-ln(X t-1 )] =100g
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4 2.1.1 – Growth Formulae 1) Percentage growth = 100[X t -X t-1 ]/X t-1 =100g 1a) [X t -X t-1 ]/X t-1 = g 1b) X t = (1+g) X t-1 1c) X t = X 0 (1+g) t 1d) ln(X t ) = ln(X 0 ) + t ln(1+g) 1e) ln(X t ) = ln(X 0 ) + gt
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5 2.1.1 – Growth Example Sales have risen from 200 to 230. Calculate growth [X t -X t-1 ]/X t-1 = g [230-200]/200 = g 0.15=g A baby elephant is born weighing 100 lbs and grows 12% every month. How much does a 10- month old weigh? X t = X 0 (1+g) t X 10 = 100 (1.12) 10 X 10 = 311
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6 2.1.1 – Growth Formulae 2) % growth = 100 [ln(X t )-ln(X t-1 )] =100g 2a) ln(X t ) – ln(X t-1 ) = g 2b) ln(X t /X t-1 ) = g 2c) X t /X t-1 = e g 2d) X t = e g X t-1 2e) X t = X 0 e gt 2f) ln(X t ) = ln(X 0 ) + gt
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7 2.1.1 – Growth Example The molecular compound L 0 RN 3 gives off radiation of e 2 radon’s per mole. If this radiation increases 5% every half-life, how much radiation will be emitted after 20 half-lives? ln(X t ) = ln(X 0 ) + gt ln(X t ) = ln(e 2 ) + 0.05(20) ln(X t ) = 2ln(e) + 1 ln(X t ) = 3 X t = e 3
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8 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Linear Time Trend: X t = a + bt Rate of growth: b/X t -straight line relationship -same increase every period -decrease if b is negative
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9 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Linear Time Trend (b>0): x=7+2t 0 5 10 15 20 25 1 2 3 4 5 6 7 8 t x x
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10 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Linear Time Trend: X t = a + bt Examples: -Bank Account: Starting balance and regular deposits -Age: Starting age +1 every 365.25 days -Mortgage payments: Principle – regular payments
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11 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Linear Time Trend (b<0): x=250,000-10,000t 0 50000 100000 150000 200000 250000 300000 1 2 3 4 5 6 7 8 t x x
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12 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Quadratic Time Trend: X t = a + bt +ct 2 Rate of growth: (b+2ct)/X t -U-shaped (c>0) or inverted U (c<0) -negative growth, then no growth, then positive or visa versa
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13 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Quadratic Time Trend c>0: x=15-10t+t*t -15 -10 -5 0 5 10 1 2 3 4 5 6 7 8 t x x
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14 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Quadratic Time Trend: X t = a - bt +ct 2 Examples: -Working out: increases mass before decreasing it (negative before positive) -Investing: Decreased disposable income now for increased in future
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15 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Quadratic Time Trend (c<0): x=15+10t-t*t 0 10 20 30 40 50 1 2 3 4 5 6 7 8 t x x
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16 2.1.1 – Simple Models of Growth – Non-Constant Growth Rates Quadratic Time Trend: X t
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