I am trying to solve the following comparative statics problem. Assume I have four choice variables, $x_{1}, x_{2}, y_{1}$ and $y_{2}$, and two parameters $t_{1}$ and $t_{2}$. I have the following minimization problem,
\begin{align*} \min_{x_{1}, x_{2}, y_{1},y_{2} \geq 0} &K_{1}x_{1} + K_{2}x_{2} + y_{1}^{-\gamma} + y_{2}^{-\gamma} \\ &s.t \hspace{3pt} y_{j} - d_{ij}x_{i} \leq d_{ij}t_{i} \hspace{3pt} \text{for} \hspace{3pt}i =1,2, j=1,2 \end{align*}
$K_{1},K_{2},\gamma, d_{ij} > 0$ are additional parameters that are not relevant to the question at hand.
I would like to understand how the optimal values of $x_{1}$ and $x_{2}$ change, when only $t_{1}$ increases.
There are all of this monotone comparative statics results, however, my intuition is that the behavior should not be monotone. $x_{1}$ should weakly decrease and $x_{2}$ should weakly increase.
Any suggestions?
I edited the constraint because it read $y_{i}$, instead of $y_{j}$.