Consumption and Portfolio Choice under Internal Multiplicative Habit Formation

Abstract

This paper explores the optimal consumption and investment behavior of an individual who derives utility from the ratio between his consumption and an endogenous habit. We obtain closed-form policies under general utility functionals and stochastic investment opportunities by developing a nontrivial linearization to the budget constraint. This enables us to explicitly characterize how habit formation affects the marginal propensity to consume and optimal stock–bond investments. We also show that in a setting that combines habit formation with Epstein–Zin utility, consumption no longer grows at unrealistically high rates at high ages and investments in risky assets decrease.

Duality methods for stochastic optimal control problems in finance

“…In fact, Schroder and Skiadas (2002) Complementary to this fundamental duality result, we develop an approximating mechanism similar to the one of Chapter 2. We test the mechanism on the approximation proposed by van Bilsen et al (2020a). This approximation is proven to be accurate under a wide variety of circumstances.…”

“…First, we provide the general evaluation mechanism and comment on related technicalities. Second, we present the approximate solution proposed by van Bilsen et al (2020a). In addition to this, we rely on the duality relation in (3.3.5) to develop a corresponding dual approximation.…”

“…Subsequently, we present the analytical approximation to optimal (ratio) consumption, c opt t , proposed by van Bilsen et al (2020a). On the grounds of our evaluation mechanism, and the relations in (3.4.3), we develop a corresponding novel dual approximation.…”

“…The motivation for such an expansion is that the habit level closely tracks optimal consumption. According to Theorem 3.1 in van Bilsen et al (2020a), this approximation is for all t ∈ [0, T ] given by:…”

“…As for the possibility of a declining standard of living, one can employ a geometric specification of the habit level, cf. Kozicki and Tinsley (2002), Corrado andHolly (2011), andvan Bilsen et al (2020a). Unlike the arithmetic habit levels, this geometric specification relies on the logarithmic transformation of consumption, and can therefore decrease over time.…”