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: $$$$U = \int_0^\infty e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] \operatorname{dt} \hspace{5mm} (1)$$$$ Agents maximize $$(1)$$ subject to the assets' restriction: $$$$\dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a$$$$ The Hamiltonian is given by $$$$\mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a ]$$$$ The maximum principle conditions are: $$\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: $$$$\mathcal{H} = e^{-(\rho-n)t} \cdot \left[ \frac{c^{1-\theta}-1}{1-\theta} \right] + \lambda \cdot [ w - c + (1-\tau)r \cdot a - n \cdot a + v]$$$$ The maximum principle conditions are: $$\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 > 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$$ in the assets's restriction, we have: \begin{align*} \dot{k} &= -c + (1-\tau) \cdot r \cdot k - n \cdot k + \tau \cdot r \cdot k \\ &= -c - (r+n) \cdot k \end{align*} and, therefore, $$\boxed{ \dot{k} = -c + (A-\delta-n) \cdot k}$$ Is what I did correct?

• What is the full name of the AK model? Is this a homework problem? – High GPA Oct 12 '20 at 22:37
• @HighGPA, it's actually only AK model. It's the class of endogenous growth models with production function given by $Y(t) = AK(t), A > 0$. It's not a homework problem. I'm just trying to properly understand how to introduce taxes in the model and if I am required to include the government in the production function if it's spending is not productive. Thanks – Pedro Cunha Oct 12 '20 at 23:41
• @PedroCunha problem with these sort of questions is that although they show a lot of work they do not really allow for high quality answer. 1. Taxes can be always introduced in various ways. It’s hard to comment on that without knowing details sometimes this is even up to discretion of the modeler. This site is not really supposed to be discussion site rather Q&A. 2. Often questions of kind are my steps correct? Without anything else are a bit of catch 22, for people potentially giving answers. Just answering it is correct is quite low quality answer. This looks more or less correct but – 1muflon1 Oct 13 '20 at 14:08
• Spending a lot of effort to verify that and in the end just post comment (as most users would avoid answering just “it’s correct”) does not create much incentives for people to go through that work. – 1muflon1 Oct 13 '20 at 14:10
• @PedroCunha it could be if you want to examine situation where government exist only for transfers of resources. – 1muflon1 Oct 13 '20 at 14:20

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)