Soft habits

Models of habit formation in consumption typically specify utility over the excess of consumption above some habit level. This specification is unsatisfactory in settings where agents occasionally have to tolerate consumption below the habit level. More importantly, they often imply infeasible solutions with realistically low riskless rates. We propose an alternative specification, where the curvature of the utility function rises steeply for consumption below the habit level, but without utility falling abruptly to minus infinity. We explore analytically the key features of the implied behavior and present representative numerical solutions of the model in continuous time. We then simulate investor-consumer behavior with these preferences and compare this behavior to simpler rules of thumb. We find that soft habits, like hard habits, imply procyclical risk taking. Soft habits also allow some smoothing, especially in the downward direction. However, its most distinguishing feature takes the form of deliberate efforts to build sufficient capital to limit the probability of consumption having to fall below habits. Because the priority given to the buildup of wealth, the question of smoothing remains mostly moot in practice. The simpler rules of thumb tend to smooth more and save less that the base case and thus lead to insufficient buildup of capital over time. Of the simpler rules, the relatively best result is found for behavior as if preferences were CRRA with risk aversion somewhere between the soft-habit risk aversion for consumption above and below the habit level.