In this book about intertemporal optimization (page 33) I've found this difference equation:
$x_{t+1}=ax_t \quad, \quad a>0$
The solution is:
$x_t=a^t x_0$
where $x_0$ is the initial value of $x_t$. Now look at the plots for $a=2, \quad x_0 \in {0.5, 1, 2} $
The values of $x_t$ on the left seem much lesser then they should be.
For $x_0 = 0.5$ we should have
$x_1 = 2^1 0.5 = 1$
$x_2 = 2^2 0.5 = 2$
$x_3 = 2^3 0.5 = 4$
But the graph is very different (the lower curve - I can't paste it here because the book is freely downloadable, but copyrighted). Or maybe my interpretation of difference equations is wrong. For t=3 we should have $x_3 = 2^3 0.5 = 4$, but the value measured on the left scale doesn't even reach 2. Am I misinterpreting difference equations or is there something wrong in the plot? Hope it's not too much a silly question.