How does one prove the convergence of a trajectory in the Ramsey model?
See Romer (2006, Sec 2.3) for details of notation.
Dynamics are given by
\begin{align}
\dot k &amp;= f(k) - (n+g)k - c\\
\dot c &amp;= \frac{c}{\theta}(f'(k) - \rho - \theta g)
\end{align}

Presume you are located somewhere on the $\dot k = 0$ locus. Now if we increase $c$ one gets
\begin{align}
\frac{\partial \dot k}{\partial c} = -1 &lt; 0
\end{align}

That is, if we are above $\dot k = 0$ capital decreaes over time. Hence at point $A$ we move to the left.

Now presume you are located somewhere on the $\dot c = 0$ locus. Now if we increase $k$ one gets
\begin{align}
\frac{\partial \dot c}{\partial k} = \frac{c}{\theta} f''(k) &lt; 0
\end{align}

That is, if we are right of $\dot c = 0$ consumption decreaes over time. Hence at point $A$ we move up in addition (left of $\dot c = 0$).