Now consider a market wherein the only asset is a bond. The bond pays off one unit of numeraire in each state.

\begin{align*} D_{sl}U^i(x^i)-\lambda^i_sp_{sl}&=0,\\ -\lambda^i_0q^1+\sum_{s=1}^S\lambda^i_s&=0,\\ \text{etc.}& \end{align*}

Any vector $\displaystyle \pi^i=\left(\frac{\lambda^i_1}{\lambda^i_0}, \ldots,\frac{\lambda^i_S}{\lambda^i_0}\right)$ is a state price vector. But since $\rank A\lt S$, state prices are not unique, and so there is no market force equilibrating marginal rates of substitution across different states of the world.

Market incompleteness creates problems for the existence of equilibrium. Radner (1973) proved the existence of equilibrium with an additional assumption that exogenously bounded the set of allowable asset positions. Hart (1975) discusses non-existence and other perversions of the Arrow-Debreu world that arise in these models. Polemarchakis (1990) has a good summary of the issues surrounding existence; in particular, how generic existence is achieved in some asset structures.

 

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