Does observational equivalence always hold in hyperbolic discounting models?

Observation equivalence holds in the hyperbolic discounting models such as Laibson (1996), Barro (1999), and Krusell et al (2002). We study a hyperbolic discounting model where the policy function cannot be replicated by a geometric discounting model. Under the logarithmic utility and Cobb-Douglas production, we obtain the explicit solution for consumer's consumption-saving decision. Different from the literatures of exponential discounting, our model shows that the habit persistence affects consumer''s consumption-saving decision. Therefore, observational equivalence does not hold in our hyperbolic discounting model.