Under the assumptions of Proposition 6.40, suppose that the dividend process (left(d_{t} ight)_{t in mathbb{N}}) is defined

Under the assumptions of Proposition 6.40, suppose that the dividend process \(\left(d_{t}\right)_{t \in \mathbb{N}}\) is defined by

\[\log d_{t+1}=\mu+\log d_{t}+\varepsilon_{t+1}\]

with \(\mu>0\) and where \(\left(\varepsilon_{t}\right)_{t \in \mathbb{N}}\) is a sequence of independent and identically distributed normal random variables with zero mean and variance \(\sigma^{2}\). Show that there exists an (intrinsic) bubble of the form \(\mathbf{u}^{\prime}\left(e_{t}\right) \beta_{t}=c d_{t}^{\lambda}\) where \(c\) is a constant and \(\lambda\) is the solution of the equation \(\lambda^{2} \sigma^{2} / 2+\lambda \mu+\log \delta=0\) (see Froot \& Obstfeld [745]).