$W = W_A + W_B,$ where $$W_A = 3/4 \ln(x_a) + 1/4 \ln(y_a),$$ and $$W_B = 2/3 \ln(x_b) + 1/3 \ln(y_b).$$ How can we maximise $W$ with given constraints $$1-) \quad e_x^a + e_x^b = x_a + x_b,$$ $$2-) \quad e_y^a + e_y^b = y_a + y_b,$$ $$3-) \quad p_x x_a + p_y y_a = p_x e_x^a + p_y e_y^a,$$ $$4-) \quad p_X x_b + p_y y_b = p_x e_x^b + p_y e_y^b,$$ $$5-) \quad p_x > 0, p_y > 0,$$ where $(e_x^a, e_y^a) = (2,1)$ and $(e_x^b, e_y^b) = (1,2)$ ?
We have the unknowns (x_a, y_a, x_b, y_b, p_x, p_y), and we need to maximise $W$, so I tried to maximise each $W_i$ to maximise $W$. To do that, I have used the Lagrange's multiplier method, but after doing lots of algebra, I couldn't find any solution to this maximisation problem.
So, first of all, how can we solve this problem ? Secondly, is there any quick way of solving this kind of "complex" maximisation problems ?
Any help or hint is appreciated.
If we sum equation $3$ and $4$, using and $1$, we can derive the equation $2$, so we have $6$ unknowns and only 5 equation if we also consider the two equations coming from Lagrange's method, hence the system should not be solvable uniquely. However, this is a problem coming from economics, and I would expect it to have a solution that maximises this function $W$.