So I have the following problem: I'm trying to get a demand function for a nonlinear 2 variable optimisation problem. There are 3 inequality constraints. Doing the usual thing I get the following results for both variables:
$$X= \frac{-P^2 - Pd\lambda + 2Pd + Pw - d\lambda w}{2P(P - d)}$$ and $$Y=\frac{d(P\lambda - P + \lambda w - w)}{2(P - d)} $$
Where $\lambda$ ,$w$ and $d$ are known parameters. That should yield a function for the demand of both $X$ and $Y$
The thing is, when I plot this, depending on the value of P, the complementary slackness conditions are violated ($Y<0$). for references, the inequality constraints are:
$$ -Y\leq 0 $$ $$ -X - 1\leq 0 $$ $$ X - 1\leq 0 $$
Which is to say $X$ lives in the interval $[-1,1]$ and $X$ is nonnegative.
I'm no expert on inequality constraints, so my question is: should I "divide" the problem into 2? say, given that $Y$ is increasing in $P$, make the restriction active until it crosses 0, then solve 2 problems before and after? Maybe I'm missing a restriction? Maybe I'm doing it all wrong by assuming an interior solution (i.e., all $\mu$ are 0)?
The whole idea is to get some non constant (hopefully decreasing), smooth function of X as the price increases.