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Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter. I have two optimization problems. The first problem (P1) is given as \begin{align} &\max_{x,y,z}f(x,y,z;a) \\ &\text{s.t. } y=z. \end{align} In the second problem (P2), I solve the unconstrained maximization, that is, \begin{equation} \max_{x,y,z}f(x,y,z;a). \end{equation}

Let the first dimension of the maximizers of these functions be $x_{1}^{*}$ and $x_{2}^{*},$ respectively. For two real numbers $l$ and $u$, I have that $a\le l \implies x_{1}^{*} \ge x_{2}^{*}$ and $a \ge u \implies x_{1}^{*} \le x_{2}^{*}$.

Is there a way for me to conjecture that there exists a single crossing property for $x^{*}$s? That is, can I claim that there exists an $\bar{a}$ such that $a\le\bar{a} \iff x_{1}^{*} \ge x_{2}^{*}$ ?

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  • $\begingroup$ To clarify, when I say "first dimension of the maximizers", I mean that for a vector of maximizers as $(x^*,y^*,z^*)$, consider only $x^*.$ $\endgroup$ – emper May 30 '15 at 18:56

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