The fundamental nature of HARA utility

Many models in Economics assume a utility function belonging to the HARA family. This paper shows that HARA utility is more fundamental to economic analysis. The HARA functional form is the unique form which satisfies basic economic principles in an optimization context. Using HARA is therefore not just a matter of convenience or tractability but rather emerges from economic reasoning, i.e., it is inherent in the economic optimization problem. The paper applies Lie symmetries to the optimality equation of Merton’s (1969, 1971) widelyused intertemporal model of the consumer-investor in order to show the inherent nature of the HARA utility function. Lie symmetries derive the conditions whereby the optimal solution remains invariant under scale transformations of wealth. The latter arise as the result of growth over time or due to the effects of policy. The symmetries place restrictions on the model, with the key one being the use of HARA utility. We show that this scale invariance of agents’ wealth implies linear optimal solutions to consumption and portfolio allocation and linear risk tolerance (and vice versa). The results have broad implications, as the model studied is a fundamental one in Macroeconomics and Finance. The paper demonstrates the use of Lie symmetries as a powerful tool to deal with economic optimization problems.