Econ. 1540 Lecture #2, D.N. DeJong

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).

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: ∆Y = slope•∆X, or ∆Y = b•∆X

Links to calculus:

• slopes are analogous to derivatives: ∂Y/∂X = b;

• total changes are measured using differentiation:

∆Y = ∂Y/∂X•∆X

Y

X

a

b

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 in X: ∆Y = ∂Y/∂X •∆X

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###### Document Summary

Mathematical background: linear equations, nonlinear equations, measuring rates of change: the logarithmic function, modeling growth: the exponential function. 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. D y = slope d x, or d y = b d x. Links to calculus: slopes are analogous to derivatives: y/ x = b, total changes are measured using differentiation: Key difference between linear and nonlinear equations: slope becomes a function of x. Change in y resulting from a change in x:

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