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.
