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ECON 1540 Chapter 2

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York University
ECON 1540
Othon Alexandrakis

Mathematical Background •Linear Equations •Nonlinear Equations •Measuring Rates of Change: the logarithmic function •Modeling growth: the exponential function Econ. 1540 Lecture #2, D.N. DeJong Linear Equations Y = a + bX a, b positive parameters a: intercept – portion of Y that is independent of X (i.e., value of Y when X=0). b: slope –response of Y to a change in X (i.e., ratio of the change in Y to the change in X –rise over run). Y a b X Key feature: slope is not a function of position on the line (i.e., not a function of the value of X). Measuring the total change in Y resulting from a given change in X: ΔY = slope•ΔX, or ΔY = b• ΔX Links to calculus: •slopes are analogous to derivativY/∂X = b; • total changes are measured using differentiation: ΔY = ∂Y/∂X•ΔX Econ. 1540 Lecture #2, D.N. DeJong Nonlinear Equations Key difference between linear and nonlinear equations: slope becomes a function of X. Generic representation: Y = f(X) Slope: ∂Y/X Change in Y resulting from a change iΔY = Y/∂X •ΔX Econ. 1540 Lecture #2, D.N. DeJong Important Nonlinear Equations c 1. Power functions: Y = a + bX Slope: ∂Y/∂ X = bcX , a function of X Example: a, b, positive, c > 1 Y a X Note: since c>1, the larger X, the steeper the slope. Recall the linear relationship Y = a + bX. This is just a special case in which c=1: ∂Y/∂X = bcX c-1= b•1•X = b. Exercises: Graph (X,Y) for 0 < c < 1, -1 < c < 0, c < -1. Econ. 1540 Lecture #2, D.N. DeJong 2. Logarithmic function: Y = ln(X) Slope: ∂ Y/∂ X = 1/X ln(X) 1 X Properties: z = a⋅x l
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