Let (left{S^{*}, D ;left(theta^{1 *}, c^{1 *} ight), ldots,left(theta^{I *}, c^{I *} ight) ight}) be a Radner

Let \(\left\{S^{*}, D ;\left(\theta^{1 *}, c^{1 *}\right), \ldots,\left(\theta^{I *}, c^{I *}\right)\right\}\) be a Radner equilibrium, in the sense of Definition 6.13. Suppose that the total supply of the \(N\) securities is normalized to \(\sum_{i=1}^{I} \theta_{0}^{i, n}=1\), for all \(n=1, \ldots, N\), and that all the agents have strictly increasing utility functions \(u^{i}: \mathbb{R}_{+} \rightarrow \mathbb{R}\). Show that in equilibrium it holds that \(\sum_{i=1}^{I} c_{t}^{i *}=\sum_{n=1}^{N} d_{t}^{n}\), for all \(t=0,1, \ldots, T\).