Introduction of an asset tax in the AK model Let's start by analyzing the family's problem with the imposition of a tax. Assuming a CRRA utility function:
\begin{equation}
    U = \int_0^\infty e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] \operatorname{dt} \hspace{5mm} (1)
 \end{equation}
Agents maximize $(1)$ subject to the assets' restriction:
\begin{equation}
        \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a 
\end{equation}
The Hamiltonian is given by
\begin{equation}
    \mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a ]
\end{equation}
The maximum principle conditions are:
$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a  \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$
Is my reasoning correct so far? Or should I have included a transfer from the government to the agents somewhere? Furthermore, whether the government spending is productive or not would also change how we deal with the tax in our AK model, right?
UPDATE: Thinking a little bit more, I've reached the conclusion that the problem should be reestated as:
Maximize $(1)$ subject to assets' restriction
$$
\dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v
$$
where $v = \frac{V}{L} = \frac{\tau \cdot r \cdot a}{L}$ is a per capita lump-sum transfer from the government to the families. Therefore, the Hamiltonian is now given by:
\begin{equation}
    \mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a + v]
\end{equation}
The maximum principle conditions are:
$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$
Using the equations above, we can find that:
$$
\frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau)r - \rho]
$$
Considering now a production function in per capita form given by
$$
y = Ak, A &gt; 0
$$
we have that the firms profits are (note that I didn't include the tax in $r$ here, per the reasoning that since $r$ enters profits as an expense, in this case it's not taxable, because it's not an asset income as in the representative family's problem)
$$
\pi = Ak - \underbrace{w}_{=\frac{\partial y}{\partial L} = 0} - \delta \cdot k - r \cdot k 
$$
which means the first-order condition of profit maximization will be given by
$$
\frac{\partial \pi}{\partial k} = 0 \implies A = f'(k) = \delta - r
$$
or
$$
r = A - \delta
$$
Therefore,
$$
\boxed{ \frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau) \cdot (A-\delta) - \rho]}
$$
Now, in equilibrium, $a = k$. If we substitute this and $v = \tau \cdot r \cdot k$ in the assets's restriction, we have:
\begin{align*}
\dot{k} &amp;= -c + (1-\tau) \cdot r \cdot k - n \cdot k + \tau \cdot r \cdot k \\ &amp;= -c - (r+n) \cdot k 
\end{align*}
and, therefore,
$$
\boxed{ \dot{k} = -c + (A-\delta-n) \cdot k}
$$
Is what I did correct?
I think your math is mostly correct but I have to admit that I am not used to AK models.
A short answer for your main question:  is it ok to introduce taxation in the model without including the government?
Answer: Yes, of course. There are overwhelming literature on dynamic or static theoretical modeling of the effect of tax policy without introducing government as an agent.
For a most famous example, see J.E. Stiglitz, Effects of wealth, income, and capital gains taxation on risk taking. Quarterly Journal of Economics, 83 (1969)
In this line of research, people almost never introduce government. You could search for the works by Arrow (1965), Cass (1972), Fishburn, Richter (1960), Tobin, Mossin (1968), Merton (1969), Russell and Smith (1970)
Also you could see the first paper published by Acemoglu.
But, as you might already explored, many recent papers include government as an agent. For example: Farhi, Emmanuel. &quot;Capital taxation and ownership when markets are incomplete.&quot; Journal of Political Economy 118.5 (2010): 908-948.