Doubt regarding Walrasian equilibrium with complements for both agents

Question

There are two goods $1,2$ and two agents $ 1,2 $. Both have the utility function $ u_{i}=\min({x_{1i},x_{2i}}) $ for agent $i$ .The endowments are $(1,3)$ and $(3,1)$ for agent $1$ and $2$ respectively. I have to solve for the Walrasian equilibrium.

I found out the demand for $x_{1}$ for both agents and at this point I tried to clear the market for $x_{1}$: (setting $p_{1}=1$ and $p_{2}=p$, I get) $\frac{1+3p}{1+p}+\frac{3+p}{1+p} = 4$ At this point I generally try to solve for the equilibrium price $p$. But In this case, Its a trivial equation. What does this mean ? What is the equilibrium price?

Sorry if I am not seeing the obvious. I am new to this and dont know much.

Answer 1

Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, price $p$ supports the allocation in which 1 consumes $\left(\frac{1+3p}{1+p}, \frac{1+3p}{1+p}\right)$ and 2 consumes $\left(\frac{3+p}{1+p}, \frac{3+p}{1+p}\right)$.