Optimal planning in habit formation models with multiple goods

In this paper we investigate a model with habit formation and two types of substitute goods. We are inspired by the classical models in e.g. Carroll et al. (Am Econ Rev 90(3):341–355, 2000) where on the contrary only one good is considered. Such a family of models, even in the case of 1 good, are difficult to study since their utility function is not concave in the interesting cases (see e.g. Bambi and Gozzi (J Math Econ 91:165–172, 2020)), hence the first order conditions are not sufficient. We are inspired by the situation in which there is a lockdown in the economy and one sector closes whereas habits develop on the second good. This more elaborate model will be the subject of future work. In the present paper, we carry out a first analysis in the case of no lockdown. We introduce and explain the model which considers two goods where one of the two is related to habit stock and where the utility function is expressed as the sum of two utility functions. For this model, we provide some first results using the dynamic programming approach. We prove that the value function is a viscosity solution of the Hamilton–Jacobi–Bellman equation, and also some results on the qualitative behaviour of the value function are furnished. Such results will form a solid ground over which a deep study of the features of the solutions can be performed.