Question

State and prove Walras' law.

Answer 1

Walras’ law states that
p1z1(p1, p2) + p2z2(p1, p2) ≡ 0.
I.e, the value of aggregate excess demand is identically zero. It means it is zero for all prices not just equilibrium prices.

Proof :- let's take two agents A & B. Add up the two agents.  first agent A. Since her demand for each good satisfies her budget constraint, we have
p1x1
A(p1, p2) + p2x2
A(p1, p2) ≡ p1ω1
A + p2ω2
A
or
p1[x1
A(p1, p2) − ω1
A] + p2[x2
A(p1, p2) − ω2
A] ≡ 0
p1e1
A(p1, p2) + p2e2
A(p1, p2) ≡ 0.
This equation says that the value of agent A’s net demand is zero. That is, the value of how much A wants to buy of good 1 plus the value of how much she wants to buy of good 2 must equal zero. (Of course the amount
that she wants to buy of one of the goods must be negative—that is, she Intends to sell some of one of the goods to buy more of the other.)
We have a similar equation for agent B:
p1[x1
B(p1, p2) − ω1
B] + p2[x2
B(p1, p2) − ω2
B] ≡ 0
p1e1
B(p1, p2) + p2e2
B(p1, p2) ≡ 0.
Adding the equations for agent A and agent B together and using the definition of aggregate excess demand, z1(p1, p2) and z2(p1, p2), we have
p1[e1
A(p1, p2) + e1
B(p1, p2)] + p2[e2
A(p1, p2) + e2
B(p1, p2)] ≡ 0
p1z1(p1, p2) + p2z2(p1, p2) ≡ 0.

Since the value of excess demands add zero then value of agents excess demand must also be zero.