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.
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?