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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 $T^*$.

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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.

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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.

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  • 1
    $\begingroup$ Sounds about right, you maximize your objective function subject to your constraint. Have you tried doing this? $\endgroup$ – user20105 Oct 21 '19 at 15:46
  • $\begingroup$ @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 $\endgroup$ – B11b Oct 21 '19 at 16:23

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