No. 1/2025: Dynamic Spending and Portfolio Decisions with an Internal Soft Habit

Abstract

We solve the Merton problem for an agent with a soft internal habit whose utility is defined over the consumption-habit ratio, and where the curvature of the utility function jumps discretely to a higher level when consumption falls below the habit. The habit evolves over time as a weighted average of past consumption, so that current consumption choices have consequences for future habits. We solve the model numerically in continuous time. Optimal behavior is characterized by extensive consumption smoothing and risk taking that is high on average but varies significantly with wealth. Withdrawal rates tend to fall below portfolio rates of return, so that wealth and consumption tend to grow exponentially over time, like in AK macro models. A lack of self-awareness, whereby the agent underestimates the rate at which habits catch up with consumption, leads to significant, though hardly huge, loss of utility. Simpler rules, such as constant withdrawal rates and equity shares over time, lead to utility losses in similar orders of magnitude. However, when withdrawal rates are held constant, risk taking must be significantly lower to be optimal in a constrained sense. We believe the results should be interesting for the management of and spending from funds like the Norwegian Government Fund Global, which motivated our study. Although our results may serve as a defense of its current regulations with a fixed equity share and (roughly) fixed withdrawal rates, our results suggest that risk taking is much too high.