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This paper studies an optimal consumption problem for a loss-averse agentwith reference to past consumption maximum. To account for loss aversion onrelative consumption, an S-shaped utility is adopted that measures thedifference between the non-negative consumption rate and a fraction of thehistorical spending peak. We consider the concave envelope of the utility withrespect to consumption, allowing us to focus on an auxiliary HJB variationalinequality on the strength of concavification principle and dynamic programmingarguments. By applying the dual transform and smooth-fit conditions, theauxiliary HJB variational inequality is solved in closed-form piecewisely andsome thresholds of the wealth variable are obtained. The optimal consumptionand investment control can be derived in the piecewise feedback form. Therigorous verification proofs on optimality and concavification principle areprovided. Some numerical sensitivity analysis and financial implications arealso presented.

Optimal consumption with loss aversion and reference to past spending maximum