Finite Horizon Portfolio Selection with Durable Goods

We study the consumption and portfolio selection problem of a finitely lived agent who derives utility from the stock of durable goods. We show that the agent's effective relative risk aversion implied by the optimal portfolio tends to decline and approaches zero, as the planning horizon approaches, whereas the agent exhibits constant effective relative risk aversion in the infinite horizon model of Hindy and Huang (1993). The existence of the stock of durable goods acts as buffer stock and induces the highly risk-tolerant attitude. We approach this problem using successive transformations. We transform our problem by applying an isomorphism proposed by Schroder and Skiadas (2002) to a singular control problem involving the choice of a monotone increasing consumption process. We next transform the problem into a dual singular control problem using the dual martingale method. We then transform the dual singular control problem into an optimal stopping problem. We analyze the variational inequality arising from the optimal stopping problem and provide an integral representation of optimal strategies