<p>In your household problem you have a typo, they maximize their utility so I think you meant <span class="math-container">$\sum_{t=0}^\infty\beta^tu(c_t, L_t)$</span>. Now their income comes from the profits of the firm <span class="math-container">$\pi$</span> their wages <span class="math-container">$w_tL_t$</span> and what they borrow-lend (not directly the production <span class="math-container">$Y$</span> or government expenditures <span class="math-container">$G$</span>) and their expenditures are consumption, paying their debts (or receiving money from what they lend yesterday) and what they pay in taxes. Obviously they only pay consumption taxes, bond taxes, and labor income taxes, since consumers don't directly invest. We usually model that investment decisions are made by firms, so investment taxes will enter the firm's problem. </p>

<p>In contrast, the social planner chooses everything so all taxes will be relevant for his problem. </p>

CC BY-SA 4.0
2019-04-17T20:45:58.183Z
2019-04-17T20:45:58.183Z
Regio
<p>In a certain economy, time is discrete with periods <span class="math-container">$t=0,1,2,...$</span>. The economy is populated by many households and identical firms. The utility of a household is:</p>

<p><span class="math-container">$\displaystyle\sum^{\infty}_{t=0}\beta^t\Big(\log c_t - \gamma L_t^{\frac{1}{\gamma}}\Big)$</span></p>

<p>where <span class="math-container">$c_t$</span> is consumption and <span class="math-container">$L_t$</span> is labor in hours worked. There are two assets: capital, <span class="math-container">$K_t$</span>, and a one-period risk-free bond <span class="math-container">$b_t$</span>.</p>

<p>Capital is rented to the firms at a rate <span class="math-container">$v_t$</span>. The law of motion for capital is <span class="math-container">$K_{t+1}=(1-\delta)K_t+I_T$</span>. <span class="math-container">$I_t$</span> is gross investment and <span class="math-container">$I_t-\delta K_t$</span> is net investment.</p>

<p>The bond, is in zero net supply and yields <span class="math-container">$r_t$</span> per period.</p>

<p>Firms are owned by households and produce output <span class="math-container">$Y_t^p=K_t^{\alpha}L_t^{1-\alpha}$</span> where <span class="math-container">$0&lt;\alpha&lt;1$</span>. Wages are <span class="math-container">$w_t$</span>.</p>

<p>The government has a sequence of public spending <span class="math-container">$\{G_t\}_{t=0}^{\infty}$</span> that will be paid with a combination of taxed bonds at a rate <span class="math-container">$\tau_t^b$</span>, taxed net investment at a rate <span class="math-container">$\tau_t^I$</span>, taxed labor income at a rate <span class="math-container">$\tau_t^w$</span>, and taxed consumption at a rate <span class="math-container">$\tau_t^c$</span>. The government cannot borrow or save, <span class="math-container">$Y_t=Y_t^p$</span>.</p>

<hr>

<p>I need to set up the firm's problem, the household's problem, and the social planner's problem but have no idea how to incorporate the tax structure.</p>

<p>For example, here is what I have for the household's problem, but I'm pretty sure it's not right:</p>

<p><span class="math-container">$\begin{aligned}
\max_{c_t, L_t, K_{t+1}} \quad &amp; \displaystyle\sum_{t=0}^{\infty} \beta^t (c_t, L_t)\\
\textrm{s.t.} \quad &amp; c_t + I_t + \tau_t = Y_t = K_t^{\alpha} L_t^{1-\alpha} + G_t\\
\end{aligned}$</span></p>

<p>I'd appreciate any hints for setting up these problems.</p>

CC BY-SA 4.0
2019-04-17T22:40:33.260Z
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iBooot
<taxation><self-study><optimization><bonds>
Neoclassical model with proportional taxes
lynak
# Neoclassical model with proportional taxes

(Asked by: username_0 on 2019-04-17)

In a certain economy, time is discrete with periods $t=0,1,2,...$. The economy is populated by many households and identical firms. The utility of a household is:

$\displaystyle\sum^{\infty}\_{t=0}\beta^t\Big(\log c\_t - \gamma L\_t^{\frac{1}{\gamma}}\Big)$

where $c\_t$ is consumption and $L\_t$ is labor in hours worked. There are two assets: capital, $K\_t$, and a one-period risk-free bond $b\_t$.

Capital is rented to the firms at a rate $v\_t$. The law of motion for capital is $K\_{t+1}=(1-\delta)K\_t+I\_T$. $I\_t$ is gross investment and $I\_t-\delta K\_t$ is net investment.

The bond, is in zero net supply and yields $r\_t$ per period.

Firms are owned by households and produce output $Y\_t^p=K\_t^{\alpha}L\_t^{1-\alpha}$ where $0<\alpha<1$. Wages are $w\_t$.

The government has a sequence of public spending $\{G\_t\}\_{t=0}^{\infty}$ that will be paid with a combination of taxed bonds at a rate $\tau\_t^b$, taxed net investment at a rate $\tau\_t^I$, taxed labor income at a rate $\tau\_t^w$, and taxed consumption at a rate $\tau\_t^c$. The government cannot borrow or save, $Y\_t=Y\_t^p$.

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I need to set up the firm's problem, the household's problem, and the social planner's problem but have no idea how to incorporate the tax structure.

For example, here is what I have for the household's problem, but I'm pretty sure it's not right:

$\begin{aligned}
\max\_{c\_t, L\_t, K\_{t+1}} \quad & \displaystyle\sum\_{t=0}^{\infty} \beta^t (c\_t, L\_t)\\
\textrm{s.t.} \quad & c\_t + I\_t + \tau\_t = Y\_t = K\_t^{\alpha} L\_t^{1-\alpha} + G\_t\\
\end{aligned}$

I'd appreciate any hints for setting up these problems.

**Tags:** taxation, self-study, optimization, bonds
> **Comments:**
> **username_1**: Please don't make more work for other people by vandalizing your posts. By posting on the Stack Exchange (SE) network, you've granted a non-revocable right, under the [CC BY-SA 3.0 license](https://creativecommons.org/licenses/by-sa/3.0) for SE to distribute that content. By SE policy, any vandalism will be reverted. If you want to know more about deleting a post, consider taking a look at: [How does deleting work?](https://meta.stackexchange.com/q/5221)  

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## 1 answers

### Answer by username_2 on 2019-04-17. Score: 1

In your household problem you have a typo, they maximize their utility so I think you meant $\sum\_{t=0}^\infty\beta^tu(c\_t, L\_t)$. Now their income comes from the profits of the firm $\pi$ their wages $w\_tL\_t$ and what they borrow-lend (not directly the production $Y$ or government expenditures $G$) and their expenditures are consumption, paying their debts (or receiving money from what they lend yesterday) and what they pay in taxes. Obviously they only pay consumption taxes, bond taxes, and labor income taxes, since consumers don't directly invest. We usually model that investment decisions are made by firms, so investment taxes will enter the firm's problem.

In contrast, the social planner chooses everything so all taxes will be relevant for his problem.


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