<p>There are <span class="math-container">$N&gt;0$</span> Households in an economy. </p>

<p>The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones. </p>

<p>In this case, the objective function is given:</p>

<p><span class="math-container">$${b \times cN \times (1-T)R(a-dT)+(1-b)\times (1-c)N\times (w+t)}\over N $$</span>
(equation 1)
where
<span class="math-container">$c$</span>=the portion of the households who are wealthy ones and <span class="math-container">$0&lt;c&lt;1$</span></p>

<p><span class="math-container">$a&gt;0$</span> = each one have a wealth stock </p>

<p><span class="math-container">$R$</span>= each one earns a gross return. </p>

<p><span class="math-container">$(1-c)N$</span> is the remaining measure of households and they earn each <span class="math-container">$w&gt;0$</span></p>

<p><span class="math-container">$0&lt;T&lt;1$</span> is tax rate. </p>

<p><span class="math-container">$t$</span>= the transfer for each labor. </p>

<p>If the ruler apply a tax, the wealthy ones do not save as determined by <span class="math-container">$d&gt;0$</span></p>

<p>The government is constrained by its balanced budget which holds </p>

<p><span class="math-container">$$T\times cN \times R(a-dT)=t\times (1-c)N$$</span></p>

<p>(equation 2)</p>

<p>I need to find optimal tax rate <span class="math-container">$T^*$</span>.</p>

<p>———</p>

<p>I cannot know how to solve it. What is objective function and what is constraint function. </p>

<p>My guess is that I should maximize the equation 1 subject to equation 2. Since the question 2 consists of equality, I should integrate equation 2 into the equation 1. And then I take derivative with respect T. </p>

<p>Is this right? </p>

<p>Please let me know your ideas. Thanks a lot. </p>

<p>————</p>

<p><strong>EDIT</strong></p>

<p><span class="math-container">$$\max_{T} {b \times cN \times (1-T)R(a-dT)+(1-b)\times (1-c)N\times (w+t)}\over N $$</span></p>

<p>Subject to <span class="math-container">$$T\times cN \times R(a-dT)=t\times (1-c)N$$</span></p>

<p>First, I insert the constraint into the objective function</p>

<p><span class="math-container">$${bcN(1-T)R(a-dT)+(1-b)(1-c)Nw+(1-b)(1-c)aNt}\over N$$</span></p>

<p><span class="math-container">$${bcN(1-T)R(a-dT)+(1-b)(1-c)Nw+(1-b)TcNR(a-dT)}\over N$$</span></p>

<p>Secondly, I re-write this equation</p>

<p><span class="math-container">$${bcNR[a-(d+a)T+dT^2]+(1-b)(1-c)Nw+(1-b)cNR(aT-dT^2)}\over N$$</span></p>

<p>Thirdly, I take its derivative with respect to <span class="math-container">$T$</span> and this derivative is equal to zero. </p>

<p><span class="math-container">$${{bcNR[-(d+a)+2dT]+(1-b)cNR(a-2dT)}\over {N} }=0$$</span></p>

<p>Then, I find the optimal <span class="math-container">$T^*$</span></p>

<p><span class="math-container">$$T^*= {{(2b-1)a+bd}\over{2d(2b-1)}}$$</span></p>

<p>Since we know that <span class="math-container">$0&lt;T^*&lt;1$</span>, </p>

<p><span class="math-container">$$0&lt; {{(2b-1)a+bd}\over{2d(2b-1)}}&lt;1$$</span></p>

<p>From this inequality, I found the condition <span class="math-container">$1/2&lt;b&lt;{{a-2d}\over {2a-3d}}$</span></p>

<p>This question refers to the Laffer curve. </p>

CC BY-SA 4.0
2019-10-21T15:46:46.110Z
Sounds about right, you maximize your objective function subject to your constraint. Have you tried doing this?
Ali
CC BY-SA 4.0
2019-10-21T16:23:52.130Z
@user20105 thank you for your reply. I have added my whole solution in the ‘EDIT’ part of the the question box. I will be happy if you please check it. Is it correct? And do you have any idea that you would like to add it. Thanks a lot
studentp
<mathematical-economics><self-study><optimization><profit-maximization><optimal-taxation>
Revenue maximization problem
studentp
# Revenue maximization problem

(Asked by: username_0 on 2019-10-20)

There are $N>0$ Households in an economy.

The government has aim to maximize a weighted average of income by imposing tax on the rich people and redistribute the tax revenue to the labor ones.

In this case, the objective function is given:

$${b \times cN \times (1-T)R(a-dT)+(1-b)\times (1-c)N\times (w+t)}\over N $$
(equation 1)
where
$c$=the portion of the households who are wealthy ones and $0<c<1$

$a>0$ = each one have a wealth stock

$R$= each one earns a gross return.

$(1-c)N$ is the remaining measure of households and they earn each $w>0$

$0<T<1$ is tax rate.

$t$= the transfer for each labor.

If the ruler apply a tax, the wealthy ones do not save as determined by $d>0$

The government is constrained by its balanced budget which holds

$$T\times cN \times R(a-dT)=t\times (1-c)N$$

(equation 2)

I need to find optimal tax rate $T^\*$.

———

I cannot know how to solve it. What is objective function and what is constraint function.

My guess is that I should maximize the equation 1 subject to equation 2. Since the question 2 consists of equality, I should integrate equation 2 into the equation 1. And then I take derivative with respect T.

Is this right?

Please let me know your ideas. Thanks a lot.

————

**EDIT**

$$\max\_{T} {b \times cN \times (1-T)R(a-dT)+(1-b)\times (1-c)N\times (w+t)}\over N $$

Subject to $$T\times cN \times R(a-dT)=t\times (1-c)N$$

First, I insert the constraint into the objective function

$${bcN(1-T)R(a-dT)+(1-b)(1-c)Nw+(1-b)(1-c)aNt}\over N$$

$${bcN(1-T)R(a-dT)+(1-b)(1-c)Nw+(1-b)TcNR(a-dT)}\over N$$

Secondly, I re-write this equation

$${bcNR[a-(d+a)T+dT^2]+(1-b)(1-c)Nw+(1-b)cNR(aT-dT^2)}\over N$$

Thirdly, I take its derivative with respect to $T$ and this derivative is equal to zero.

$${{bcNR[-(d+a)+2dT]+(1-b)cNR(a-2dT)}\over {N} }=0$$

Then, I find the optimal $T^\*$

$$T^\*= {{(2b-1)a+bd}\over{2d(2b-1)}}$$

Since we know that $0<T^\*<1$,

$$0< {{(2b-1)a+bd}\over{2d(2b-1)}}<1$$

From this inequality, I found the condition $1/2<b<{{a-2d}\over {2a-3d}}$

This question refers to the Laffer curve.

**Tags:** mathematical-economics, self-study, optimization, profit-maximization, optimal-taxation

> **Comments:**
> **username_1**: Sounds about right, you maximize your objective function subject to your constraint. Have you tried doing this?  
> **username_0**: @user20105 thank you for your reply. I have added my whole solution in the ‘EDIT’ part of the the question box. I will be happy if you please check it. Is it correct? And do you have any idea that you would like to add it. Thanks a lot  

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## 0 answers

