In the context of Sect. 6.4, let (g(t, tau):=left(e_{t+tau} / e_{t} ight)^{1 / tau}) and suppose that

In the context of Sect. 6.4, let \(g(t, \tau):=\left(e_{t+\tau} / e_{t}\right)^{1 / \tau}\) and suppose that the representative agent has a power utility function of the type

\[\mathbf{u}(x)=\frac{x^{1-\gamma}}{1-\gamma}\]

Show that the following relation holds in an approximate form:

\[\log (1+i(t, t+\tau))=-\log \delta+\gamma \mathbb{E}\left[\log g(t, \tau) \mid \mathscr{F}_{t}\right]\]

for all \(t \in\{0,1, \ldots, T-1\}\) and \(\tau \in\{0,1, \ldots, T-t\}\).