$$W_{t+1} = (W_t - C_t) R_{c, t+1}$$
Here $$R_{c, t+1}$$ is the 'return to the consumption asset'. What does this mean?
On the previous page (introducing equation 2.2) the authors state that "consumption asset" means "wealth portfolio", so $$R_{c,t+1}$$ is the return to the wealth portfolio, which makes sense from the equation you quote.
the log return to the consumption asset (wealth portfolio), $$r_{c,t+1}$$.