Internal habits formation and optimality

In a very influential model with internal habits, Carroll et al., (2017, 2000), establish that an increase in economic growth may cause a positive change in savings. The optimality of this result, and of many other contributions using a similar framework, has been questioned by some authors who have observed that the parametrization used in these models always implies a utility function not jointly concave in consumption and habits. In this paper, we revisit the optimality issue and, using advanced techniques in Dynamic Programming, we answer the following long-standing open questions: (i) Is the solution found in Carroll et al., (2017, 2000) optimal? (ii) Is it also unique or do other optimal solutions exist?