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In this paper we use a dynamic programming approach to analytically solve anendogenous growth model with internal habits where the key parametersdescribing their formation, namely the intensity, persistence and lag structure(or memory), are kept generic. Then we show that external and internal habitslead to the same closed loop policy function and then to the same (Pareto)optimal equilibrium path of the aggregate variables when the utility functionis subtractive nonseparable. The paper uses new theoretical results from thosepreviously developed by the dynamic programming literature applied to optimalcontrol problems with delay and it extends the existing results on theequivalence between models with internal and external habits to the case offinite memory.

On the equivalence of internal and external habit formation models with finite memory