Non-trivial steady state Consider the growth model with inelastic labor supply, full depreciation, log utility and CRS technology with the Bellman equation be defined as follows:
$$V(k)=\max(log(k^\alpha-k')+\beta V(k'))$$
st $$k\geq0\ \text{and}\ \theta k^\alpha-k'\geq0$$
As a guess I have used the usual $V(k)=a+bln(k)$ substituted on the Bellman and have derived $k'=\frac{k^\alpha\beta b}{1+\beta b}$. From this I have found the $k_{ss}=(\frac{1+\beta b}{\beta b})^{1/(\alpha-1)}$ called the non-trivial SS
This is the first SS while the other is $k_{ss}=0$
My question is how can we use the policy function to show that the system converges to the non-trivial steady steady state given any $k_0 &gt; 0$
Let's guess that the value function is of the form $a + b \ln(k)$.
Then substituting for $V(k) = a + b \ln(k)$ in the Bellman equation gives:
$$
a + b \ln(k) = \max_{k'}\left(\ln(k^\alpha - k') + \beta(a + b \ln(k')\right)
$$
The first order condition is given by:
$$
\begin{align*}
&amp;\frac{-1}{k^\alpha - k'} + \beta b \frac{1}{k'} = 0,\\
\to &amp; k' = \beta b (k^\alpha - k'),\\
\to &amp; k' = \frac{\beta b}{1+ \beta b} k^\alpha
\end{align*}
$$
If we plug this  into the objective function of the Bellman equation, we obtain the following identity:
$$
\begin{align*}
a + b \ln(k) &amp;= \ln\left(k^\alpha - \frac{\beta b}{1 + \beta b}k^\alpha\right) + \beta\left(a + b \ln\left(\frac{\beta b}{1 + \beta b}k^\alpha\right)\right),\\
&amp;= (\alpha + \beta b \alpha) \ln(k) + \ln\left(1 - \frac{\beta b}{1 + \beta b}\right) + \beta a + \beta b \ln\left(\frac{\beta b}{1 + \beta b}\right)
\end{align*}
$$
As this holds for all $k (&gt; 0)$ we can equate coefficients on both sides:
$$
\begin{align*}
a &amp;= \ln\left(\frac{1}{1 + \beta b}\right) + \beta a + \beta b \ln\left(\frac{\beta b}{1 + \beta b}\right),\\
b &amp; = \alpha + \beta b \alpha
\end{align*}
$$
The second one gives a closed form expression for $b$:
$$
b = \frac{\alpha}{1 - \beta \alpha}.
$$
Then substituting this into the first order condition gives:
$$
\begin{align*}
k_{t+1} &amp;= \frac{\beta \frac{\alpha}{1 - \beta \alpha}}{1 + \beta \frac{\alpha}{1 - \beta \alpha}}k_t^\alpha,\\
&amp;= \beta \alpha k^\alpha_t \tag{1}
\end{align*}
$$
This shows that:
$$
k_{t + 1} &gt; k_t \iff \beta \alpha k_t^\alpha &gt; k_t \iff k_t &lt; (\beta \alpha)^{\frac{1}{1 - \alpha}}
$$
So the capital stock will rise as long as $k_t$ is below $(\beta \alpha)^{\frac{1}{1 - \alpha}}$ and it will decrease if $k_t$ it is above this threshold.
According to the dynamic equation (1) above, it would appear that $k = 0$ is also a steady state. However, for $k = 0$, the first order conditions are not satisfied and in fact the value function does not exist. Anyway, for $k_t$ very close, it's value will be below $(\beta \alpha)^{\frac{1}{1 - \alpha}}$ so the stock of capital should increase to the unique steady state.