I have a two dimensional optimization problem of the form $$ v = \max_{x,y} f(x,y)+g(x,y) $$ Both $g,f$ are concave and continuously differentiable. Assume the solution can be reached by first order conditions and $(x^*,y^*)$ is unique.
And $v>0$, $g(x^*,y^*)<0$.
Now I modify the problem to $$ u = \max_{x,y} f(x,y) $$ s.t. $$ g(x,y)=0 $$
Then, following the lagrangian method, I need to solve a new problem $$ \max \mathcal{L} = f(x,y)+\lambda g(x,y) $$
My question is, is there any conclusive property regarding the value of multiplier $\lambda$? Say whether it is greater than 1 or not?
Thanks!