Econ 299 Chapter2b - 2.1.4 Growth Models The most common...

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1 2.1.4 – Growth Models The most common formulas to measure growth are: 1) [{X t -X t-1 }/X t-1 ] X 100 2) [ln(X t )-ln(X t-1 )] X 100 3) [{dX/dt}/X] X 100 4) [dln(X)/dt] X 100 -each formula has advantages and disadvantages given situation and available data
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2 2.1.4 – Growth Examples Find growth of X when ln(X) = a +bt Using 4: g = [dln(X)/dt] X 100 = 100b% Using 2: g = [ln(X t )-ln(X t-1 )] X 100 = a+bt –[a+b(t-1)] X 100 = 100b%
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3 2.1.4 – Growth Examples Find growth of X when X = a + bt Using 3: g = [{dX/dt}/X] X 100 = 100b/X % Using 1: g = [{X t -X t-1 }/X t-1 ] X 100 = {a+bt –[a+b(t-1)]}/X X 100 = 100b/X%
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4 2.1.4 – Growth Examples Find growth of X when ln(X) = a + bln(t) Using 4: g = [dln(X)/dt] X 100 = ? Using 3 (Chain Rule): g = [{dX/dt}/X] X 100 Let u = ln(x) du/dx=? Need more calculus rules.
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5 2.A – More Derivatives 1) Natural Logs If y=ln(x), y’ = 1/x -chain rule may apply If y=ln(x 2 ) y’ = (1/x 2 )2x = 2/x
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6 2.A – More Derivatives 2) Trig. Functions If y = sin (x), y’ = cos(x) If y = cos(x) y’ = -sin(x) -Use graphs as reminders
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7 2.A – More Derivatives Remember that derivatives reflect slope: Sine(blue) and Cosine(red) -1.5 -1 -0.5 0 0.5 1 1.5 1 3 5 7 9 11 13 15 17 19 21 23 25 27 x sin(x),cos(x)
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8 2.A – More Derivatives 2) Trig. Functions – Chain Rule If y = sin 2 (3x+2), y’= 2sin(3x+2)cos(3x+2)3 Try these: y=ln(2sin(x) -2cos 2 (x-1/x)) y=sin 3 (3x+2)ln(4x-7/x 3 ) 5 y=ln([3x+4]sin(x)) / cos(12xln(x))
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9 2.A – More Derivatives 2) Exponents If y = b x y’ = b x ln(b) Therefore If y = e x y’ = e x
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10 2.A – More Derivatives 2) Exponents and chain rule If y = b kx y’ = b kx ln(b)k Or more explicitly: If y = b g(x) y’ = b g(x) ln(b) X dg(x)/dx
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11 2.A – More Derivatives 2) Exponents and chain rule If y = 5 2x y’ = 5 2x ln(5)2 Or more complicated: If y = 5 sin(x) y’ = 5 sin(x) ln(5) * cos(x)
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12 2.1.4 – Growth Examples Find growth of X when ln(X) = a + bln(t) Using 4: g = [dln(X)/dt] X 100 = 100 b/t
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13 2.1.4 – Growth Examples Find growth of X when ln(X) = a + bln(t) Using 3: g = [{dX/dt}/X] X 100 = 100b[dX/dln(x) * dln(t)/dln(t) * dln(t)/dt]/ X = 100[X * b * 1/t]/X = 100b/t %
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14 2.2 Mathematical Models of Economic Relationships Example 1 – Consumption Function -consumption is based on income -even with zero income, some consumption (autonomous consumption) occurs -as income rises, consumption rises -out of every new dollar earned, a fraction, the marginal propensity to consume (mpc) is spent on consumption – remainder is saved -the mpc determines the slope of the graph
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15 2.2 Mathematical Models of Economic Relationships Consumption Function – constant slope = mpc Linear Consumption Function 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 800 900 Income Consumption
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16 2.2 Mathematical Models of Economic Relationships Consumption Function – decreasing slope = mpc Quadratic Consumption Function 0 50 100 150 200 250 300 350 400 450 0 100 200 300 400 500 600 700 800 900 Income Consumption
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17 2.2 Mathematical Models of Economic Relationships Consumption Function – slope = mpc Consumption = 100+0.5income Mpc = dc/di = 0.5 Consumption = 100+0.5income-0.02income 2 Mpc = dc/di = 0.5-0.04income Are any other functional forms viable for consumption?
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18 2.2 Mathematical Models of Economic Relationships Example 2 – Short-run Phillips Curve -if no excess demand in the economy, the economy will be at the natural rate of unemployment
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