I have the following utility maximization problem with inequality constraints:
Objective function given by $U(x_1,x_2)=\ln(x_1)+\beta \ln(x_2)$ where $0<\beta<1$, and the constraints are given by $0\leq w_1 - x_1$ and $0\leq w_1+w_2-x_1-x_2$, where $w_1$ and $w_2$ are strictly positive.
Because the objective function and constraint functions are concave and differentiable, we can use Kuhn-Tucker and find the $(x_1, x_2)$ pair that solves the first-order conditions of the Lagrangian.
With the Lagrangian expressed as: $$L=U(x_1, x_2)+\lambda_1(w_1-x_1)+\lambda_2(w_1+w_2-x_1-x_2),$$
I have determined that $\lambda_1=\frac{1}{w_1}-\frac{\beta}{w_2}$ and $\lambda_2=\frac{\beta}{w_2}$. My question is, how should I interpret the value of $\lambda_2$?
My current belief is that it represents the increase in the maximum utility when $w_1+w_2$ increases by $1$, but I'm confused by the fact that if $w_1$ were to increase, it would also involve $\lambda_1$, unless only $w_2$ increased. Can someone help to clarify?
With $\beta > 0$ the maximum will be located at the boundary. So the potential maximum point is $w_1,w_2$. At this point we have
$$ \nabla (w_1-x_1) = (-1,0)\\ \nabla (w_1+w_2-x_1-x_2) = (-1,-1)\\ \nabla (\ln x_1 + \beta\ln x_2) = \left(\frac{1}{w_1},\frac{\beta}{w_2}\right) $$
so the condition for a maximum is the existence of $\lambda_1\ge 0 ,\lambda_2 \ge 0$ such that $$ \lambda_1 (-1,0) + \lambda_2 (-1,-1) = - \left(\frac{1}{w_1},\frac{\beta}{w_2}\right) $$