Optimal decisions on the instantaneous rate of growth of consumption in excess of habit and money demand

Objective: this paper develops a time-continuous deterministic model of a rational consumer that maximizes his utility of the instantaneous rate of growth of consumption in excess of habit and real money balances. Methodology: we introduce the concept of utility of the instantaneous rate of growth of consumption in excess of habit. The necessary conditions for an interior solution lead to a Bernoulli’s differential equation of second order. Results: closed form solutions of the instantaneous rate of growth of consumption in excess of habit, consumption, and real money balances are provided. Moreover, economic welfare of the consumer is computed and comparative statics exercises are carried out. Novelty: this the first time that in the utility function is included the growth rate of consumption in excess of habit, which provides richer environments for studying rational behavior of habit formation. Conclusions: under the assumption of logarithmic utility, it is found that consumption increases linearly with time due to the habit, while money demand remains constant to finance the habit through time.