\begin{equation*} \frac{D_{sl}U^i(x^i)}{D_{sm}U^i(x^i)}=\frac{p_{sl}}{p_{sm}} \end{equation*}

Marginal rates of substitution between goods in the same state are equal across individuals.

\begin{align*} \frac{D_{sl}U^i(x^i)}{D_{tm}U^i(x^i)}&=\frac{\lambda^i_s}{\lambda^i_t} \frac{p_{sl}}{p_{tm}}\\[6pt] \frac{\lambda^i_s}{\lambda^i_t}=\frac{q_s}{q_t} \end{align*}

Marginal rates of substitution between goods in different states equal a constant times the marginal rate of substitution between the numeraires in the different states. With complate markets they are also equal.