<p>I think your math is mostly correct but I have to admit that I am not used to AK models.</p>
<p>A short answer for your main question:  is it ok to introduce taxation in the model without including the government?</p>
<p>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.</p>
<p>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)</p>
<p>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)</p>
<p>Also you could see the first paper published by Acemoglu.</p>
<p>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.</p>

CC BY-SA 4.0
2020-10-14T03:33:34.650Z
Is there anything obviously wrong? Thank you for answering, @High GPA
Pedro Cunha
CC BY-SA 4.0
2020-10-13T21:03:08.073Z
2020-10-14T05:39:59.603Z
2020-10-14T05:39:59.603Z
High GPA
<p>Let's start by analyzing the family's problem with the imposition of a tax. Assuming a CRRA utility function:
<span class="math-container">\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}</span>
Agents maximize <span class="math-container">$(1)$</span> subject to the assets' restriction:
<span class="math-container">\begin{equation}
        \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a 
\end{equation}</span>
The Hamiltonian is given by
<span class="math-container">\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}</span>
The maximum principle conditions are:
<span class="math-container">$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a  \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$</span>
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?</p>
<p>UPDATE: Thinking a little bit more, I've reached the conclusion that the problem should be reestated as:</p>
<p>Maximize <span class="math-container">$(1)$</span> subject to assets' restriction
<span class="math-container">$$
\dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v
$$</span>
where <span class="math-container">$v = \frac{V}{L} = \frac{\tau \cdot r \cdot a}{L}$</span> is a per capita lump-sum transfer from the government to the families. Therefore, the Hamiltonian is now given by:
<span class="math-container">\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}</span>
The maximum principle conditions are:
<span class="math-container">$$\begin{cases}
    \frac{\partial \mathcal{H}}{\partial c} = 0 \implies e^{-(\rho-n)t} \cdot c^{-\theta} = \lambda \\ \dot{a} = \frac{\partial \mathcal{H}}{\partial \lambda} \implies \dot{a} = w - c + (1-\tau)r \cdot a - n \cdot a + v \\ \dot{\lambda} = -\frac{\partial \mathcal{H}}{\partial a} \implies \dot{\lambda} = -\lambda \cdot [(1-\tau)r - n] \\ \lim_{t \to \infty}  [\lambda \cdot a] = 0
\end{cases}
$$</span>
Using the equations above, we can find that:
<span class="math-container">$$
\frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau)r - \rho]
$$</span>
Considering now a production function in per capita form given by
<span class="math-container">$$
y = Ak, A &gt; 0
$$</span>
we have that the firms profits are (note that I didn't include the tax in <span class="math-container">$r$</span> here, per the reasoning that since <span class="math-container">$r$</span> 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)
<span class="math-container">$$
\pi = Ak - \underbrace{w}_{=\frac{\partial y}{\partial L} = 0} - \delta \cdot k - r \cdot k 
$$</span>
which means the first-order condition of profit maximization will be given by
<span class="math-container">$$
\frac{\partial \pi}{\partial k} = 0 \implies A = f'(k) = \delta - r
$$</span>
or
<span class="math-container">$$
r = A - \delta
$$</span>
Therefore,
<span class="math-container">$$
\boxed{ \frac{\dot{c}}{c} = \frac{1}{\theta} \cdot [(1-\tau) \cdot (A-\delta) - \rho]}
$$</span>
Now, in equilibrium, <span class="math-container">$a = k$</span>. If we substitute this and <span class="math-container">$v = \tau \cdot r \cdot k$</span> in the assets's restriction, we have:
<span class="math-container">\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*}</span>
and, therefore,
<span class="math-container">$$
\boxed{ \dot{k} = -c + (A-\delta-n) \cdot k}
$$</span>
Is what I did correct?</p>

CC BY-SA 4.0
2020-10-12T22:37:18.530Z
What is the full name of the AK model? Is this a homework problem?
High GPA
CC BY-SA 4.0
2020-10-12T23:41:49.060Z
@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
CC BY-SA 4.0
2020-10-13T14:08:20.277Z
@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
CC BY-SA 4.0
2020-10-13T14:10:28.787Z
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
CC BY-SA 4.0
2020-10-13T14:16:59.583Z
I'm so sorry, what you say makes complete sense, @1muflon1. I ended up updating my answer too much, leaving little for discussion. I guess my main question is: is it ok to introduce taxation in the model without including the government in the production function?
Pedro Cunha
CC BY-SA 4.0
2020-10-13T14:20:58.650Z
@PedroCunha it could be if you want to examine situation where government exist only for transfers of resources.
1muflon1
<taxation><dynamic-optimization><endogenous-growth>
Introduction of an asset tax in the AK model
Pedro Cunha
# Introduction of an asset tax in the AK model

(Asked by: username_0 on 2020-10-11)

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
$$
y = Ak, A > 0
$$
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?

**Tags:** taxation, dynamic-optimization, endogenous-growth

> **Comments:**
> **username_1**: What is the full name of the AK model? Is this a homework problem?  
> **username_0**: @username_1, 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  
> **username_2**: @username_0 problem with these sort of questions is that although they show a lot of work they do not really allow for username_1 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  
> **username_2**: 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.  
> **username_0**: I'm so sorry, what you say makes complete sense, @username_2. I ended up updating my answer too much, leaving little for discussion. I guess my main question is: is it ok to introduce taxation in the model without including the government in the production function?  
> **username_2**: @username_0 it could be if you want to examine situation where government exist only for transfers of resources.  

---

## 1 answers

### Answer by username_1 on 2020-10-13. Score: 2

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. "Capital taxation and ownership when markets are incomplete." Journal of Political Economy 118.5 (2010): 908-948.

> **Comments:**
> **username_0**: Is there anything obviously wrong? Thank you for answering, @username_1  

---

