Inconsistent Investment and Consumption Problems

Abstract

In a traditional Black-Scholes market we develop a verification theorem for a general
class of investment and consumption problems where the standard dynamic programming
principle does not hold. The theorem is an extension of the standard Hamilton-Jacobi-
Bellman equation in the form of a system of non-linear differential equations. We derive
the optimal investment and consumption strategy for a mean-variance investor without
pre-commitment endowed with labor income. In the case of constant risk aversion it turns
out that the optimal amount of money to invest in stocks is independent of wealth. The
optimal consumption strategy is given as a deterministic bang-bang strategy. In order to
have a more realistic model we allow the risk aversion to be time and state dependent. Of
special interest is the case were the risk aversion is inversely proportional to present wealth
plus the financial value of future labor income net of consumption. Using the verification
theorem we give a detailed analysis of this problem. It turns out that the optimal amount
of money to invest in stocks is given by a linear function of wealth plus the financial value
of future labor income net of consumption. The optimal consumption strategy is again
given as a deterministic bang-bang strategy. We also calculate, for a general time and state
dependent risk aversion function, the optimal investment and consumption strategy for a
mean-standard deviation investor without pre-commitment. In that case, it turns out that
it is optimal to take no risk at all.

Keywords: Time consistency; time inconsistency; stochastic control; dynamic programming; pseudo Hamilton-Jacobi-Bellman equation; mean-variance; mean-standard deviation; state dependent risk aversion.

JEL Classification: C61