•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).
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
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)
Change in Y resulting from a change iΔY = Y/∂X •ΔX
Econ. 1540 Lecture #2, D.N. DeJong Important Nonlinear Equations
1. Power functions: Y = a + bX
Slope: ∂Y/∂ X = bcX , a function of X
Example: a, b, positive, c > 1
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
z = a⋅x l