Comparative Static on Min of Sum of Two Convex Functions

Question

Let $g_1(x)$ and $g_2(x)$ be convex functions from $\mathbb{R}\rightarrow \mathbb{R}$, and assume that the global minimum of each $g_i$, denoted $x_i^*$, is unique, with $x_1^*<x_2^*$. Let $x^*(a)$ denote the minimum of $$ (1-a) g_1(x) + ag_2(x)$$ for $a \in [0,1]$. We know $x^*(a) \in [x_1^*,x_2^*]$. Is it possible to show that $dx^*(a)/da >0$?

Answer 1

Let $x(a) =\min _{x}\{(1 -a)g_{1}(x) +ag_{2}(x)\}$ and $\sigma (a) =\arg \min _{x}\{(1 -a)g_{1}(x) +ag_{2}(x)\}$.

Then \begin{equation}\frac{ \partial x(a)}{ \partial a} = -g_{1}(\sigma (a)) +g_{2}(\sigma (a)) \end{equation}So for $a =0$ your claim is true but generally it does not seem to be the case.