# Optimal commodity taxation - solving for the optimal tax rates?

I need to solve for optimal tax rates for two products.


In my model there are N identical households that maximize their utility function subject to a budget constraint.U is positive utility function, D is disutility function. Y is income.

$$U(x_1,x_2,y_1,y_2)- D(x_1,x_2,y_1,y_2)$$

Subject to: $$q_1^x x_1+ q_2^x x_2+q_1^y y_1+q_2^y y_2 = Y$$

The state government wants to tax goods $x_1$ and $x_2$ and they need to solve for the optimal tax rates $t_1^x$ and $t_2^x$. $y_1$ is same good as $x_1$ but purchased in another state, $y_2$ is same good as $x_2$ but purchased from another state. State government wants to maximize the sum of all utilities, subject to raising a certain amount of revenue.

$$max ∑_hU(x_1,x_2,y_1,y_2)- D(x_1,x_2,y_1,y_2) + a[Y-q_1^x x_1- q_2^x x_2-q_1^y y_1- q_2^y y_2]$$ by choosing $t_1^x$ and $t_2^x$

Subject to $$t_1^x ∑_hx_1 + t_2^x ∑_hx_2 ≥R$$

When I take the first order conditions for $t_1^x$ and $t_2^x$, I end up with two complex equations that have Lagrange multipliers in them, therefore I am not able to obtain expressions for each tax rate that would not have the Lagrange multipliers in them. Do I need to make more assumptions in order to obtain the expressions?

• Do you have a specific functional form for your utility functions? – jmbejara Mar 7 '15 at 21:36
• Why do you have a disutility function? The utility function is only one, and it is possible to have inputs that diminishes the utility, they are called "bads" - so don´t worry, and drop the desutility function - it doesn´t exist. Now you have a normal lagrangian and can calculate it with no problems. – Joao Luiz Pacheco Jan 26 '16 at 15:45
• I still believe that my comment is the answer, look, a Lagrangian has a objective function and restrictions. You are trying to use two objective functions where you can use only one. If this does not solve, then show your fisrt order conditions to make the problem more explicit. – Joao Luiz Pacheco Jan 26 '16 at 21:41