I am struggling with a constrained maximisation problem in welfare economics. The question stems from my personal project and somehow proved very difficult despite simple look. The problem goes as follows
Suppose there are 2 consumers, whose valuation functions $v_1(x_1), v_2(x_2)$ are concave and increasing. There is a single producer whose cost function $c(y)$ is convex and increasing. In general, $x_1,x_2,y$ are vectors in $R^n_+$ representing amount of consumptions and production. Normalize $v_1(0)=v_2(0)=c(0)=0$.
Define Welfare $$W=\max_{x_1,x_2,y\in R_+^n}\{v_1(x_1)+v_2(x_2)-c(y)\}$$ $$\mathrm{s.t.}\quad x_1+x_2=y$$ Define Welfare without consumer 1 $$W_{-1}=\max_{x_2,y\in R_+^n}\{v_2(x_2)-c(y)\}$$ $$\mathrm{s.t.}\quad x_2=y$$ Define Welfare without consumer 2 $$W_{-2}=\max_{x_1,y\in R_+^n}\{v_1(x_1)-c(y)\}$$ $$\mathrm{s.t.}\quad x_1=y$$ Prove that Welfare exhibits substitution between consumer 1 and 2, that is, $$W_{-1}+W_{-2}\geq W$$
Implicitly, we also assume these welfare maximisation problems are well-defined. Feel free to add additional assumption if needed. I managed to prove such inequality when $n=1$, but failed to prove when $n\geq 2$. I simulated various simulation results in Mathematica when $n=2$ and there is not a single one disproving the inequality.