An Isomorphism between Asset Pricing Models with and without Linear Habit Formation

We show an isomorphism between optimal portfolio selection or competitive equilibrium models with utilities incorporating linear habit formation, and corresponding models without habit formation. The isomorphism is expressed through an explicit transformation of consumption plans, utilities, endowments, state prices, wealth processes, security prices, and trading strategies that can be used to mechanically transform known solutions not involving habit formation to corresponding solutions with habit formation. For example, the Constantinides (1990) and Ingersoll (1992) solutions are mechanically obtained from the familiar Merton solutions for the additive utility case, without recourse to a Bellman equation or first order conditions. More generally, recent solutions to portfolio selection problems with recursive utility and a stochastic investment opportunity set are readily transformed to novel solutions of corresponding problems with utility that combines recursivity with habit formation. Our methodology also applies in the context of Hindy-Huang-Kreps preferences, where our isomorphism shows that the solution obtained by Hindy and Huang (1993) can be mechanically transformed to Dybvig's (1995) solution to the optimal consumption-investment problem with consumption ratcheting.