Under the assumptions of Sect. 6.5, suppose that the representative agent is risk neutral and consider the following securities in an infinite time horizon:

(i) a security paying a constant dividend stream \(d_{t}=\bar{d}\) at all dates \(t=0,1,2, \ldots\) (in an infinite horizon). Show that the fundamental value of this security is given by \(s_{t}^{*}=\frac{\delta}{1-\delta} \bar{d}\), for all \(t \in \mathbb{N}\);

(ii) a security paying a constant dividend stream \(d_{t}=\bar{d}(1)\). At some date \(t_{0}\), a new dividend policy is unexpectedly being announced to start at the future date \(\bar{t}>t_{0}\). Such a new policy consists in the distribution of the constant dividend

\(d_{t}=\bar{d}(2)\). Show that the fundamental value of this security is given by \[s_{t}^{*}= \begin{cases}\frac{\delta}{1-\delta} \bar{d}(1), & \text { if } t