Problems
1.1
a)
\(Z(t) = X(t)Y(t)\)
The growth rate of a variable is the derivative of its natural log:
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t)Y(t))}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{d(ln(X(t))+ln(Y(t))}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dt} + \frac{dln(Y(t))}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dt} \frac{dX(t)}{dX(t)} + \frac{dln(Y(t))}{dt} \frac{dY(t)}{dY(t)}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dX(t)} \frac{dX(t)}{dt} + \frac{dln(Y(t))}{dY(t)} \frac{dY(t)}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{\dot{X}}{X} + \frac{\dot{Y}}{Y}\)
b)
\(Z(t) = \frac{X(t)}{Y(t)}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(\frac{X(t)}{Y(t)})}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{d(ln(X(t))-ln(Y(t))}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dt} - \frac{dln(Y(t))}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dt} \frac{dX(t)}{dX(t)} - \frac{dln(Y(t))}{dt} \frac{dY(t)}{dY(t)}\)
\(\frac{\dot{Z}}{Z} = \frac{dln(X(t))}{dX(t)} \frac{dX(t)}{dt} - \frac{dln(Y(t))}{dY(t)} \frac{dY(t)}{dt}\)
\(\frac{\dot{Z}}{Z} = \frac{\dot{X}}{X} - \frac{\dot{Y}}{Y}\)
1.2
a)