Consider a functional form:
$y_t$ = a$x_t$ + b ...(1)
Where, $x_t$ grows at rate g ie. $(x_{t+1} - x_t)/x_t$ = g
Find the growth rate of y?
Solution: Differentiating (1) wrt to time t
$\dot{y_t}$ = a$\dot{x_t}$
Dividing both sides by $y_t$
$\frac{\dot{y_t}}{y_t}$ = $\frac{\dot{ax_t}}{ax_t +b}$
My problem is how to bring in g into the above equation so that growth of y can be expressed in terms of growth of x?
If $\,\,\displaystyle \frac{x_{t+1}-x_t}{x_t}=g$, then $x_{t+1}=(1+g)x_t$, and so $y_{t+1}=a(1+g)x_t+b$.
Then, $\displaystyle \frac{y_{t+1}-y_t}{y_t}=\frac{agx_t}{ax_t+b}$. Due to $b$, the growth rate isn't expressible solely as a function of $g$.