What's the most straight-forward way to prove Walras's Law?

Question

Walras' Law states that summation of pi Ei(p) = 0 for all pi. We define Ei(p) = xi(p) - qi(p) - Ri. What are the next steps that I should take?

Answer 1

Let $i$ denote an agent; $j$ denote the good.

Walras' law: $p.e(p)=0$ for all $p$.

Start with the budget constraint:

$\sum_{j} p_j.x_{ij}=\sum_{j} p_j.w_{ij}$ where $w_{ij}$ is $i$'s endowment of good $j$, $x_{ij}$ is $i$'s consumption of good $j$.

In other words, $\sum_{j} p_{j}.e_{ij}=0$, where $e_{ij}=x_{ij}-w_{ij}$.

Now just add over all agents $i$. You get $\sum_{j}p_j.e_j=0$, where $e_j=\sum_i e_{ij}$ for each $j$. This is Walras' Law. Note that this applies to ALL $p$ - regardless of whether it's the equilibrium price.