Complete Markets

Theorem. Suppose that $\rank A=S$.

1. Suppose Radner prices are $(p,q)$ with state prices $\pi$, and define $\phi_0=p_0$, and for $s\geq 1$, $\phi_{sl} =\pi_sp_{sl}/p_{s1}$. Then $(x,z)\in B_R(p.q.\omega^i)$ iff $x\in B_{AD}(p,\omega^i)$.
2. Suppose Arrow-Debreu prices are $\phi$, and define $p_0=\phi_0$, $\pi_s=\phi_{s1}$, $p_{sl}=\phi_{sl}/\phi_{s1}$, and $q=\pi A$. Then $x\in B_{AD}(\phi,\omega^i)$ iff there is a $z$ such that $(x,z)\in B_R(p,q,\omega^i)$.

This fact is due to Arrow (1952/1964) — Arrow's “other theorem”. The proof is simple algebra.

Implications of Complete Markets:

• When markets are complete, Radner and Arrow-Debreu equilibrium commodity allocations are identical.
• Prices of any one type of equilibrium can be derived from the other.
• All Radner equilibria are Pareto-optimal.