Optimal Consumption and Sale Strategies for a Risk Averse Agent

In this article we consider an optimal consumption/optimal portfolio problem in which an agent with constant relative risk aversion seeks to maximize expected discounted utility of consumption over the infinite horizon, in a model comprising a risk-free asset and a risky asset in which the risky asset can only be sold and not bought. The problem is an extension of the Merton problem and a special case of the transaction costs model of Constantinides–Magill and Davis–Norman. Via various transforms we are able to make considerable progress towards an analytical solution. The solution can be expressed via a first crossing problem for an initial-value, first order ODE. The fact that we have a relatively explicit solution means we are able to consider the comparative statics of the problem. There are some surprising conclusions, such as consumption rates are not monotone increasing in the return of the asset, nor are the certainty equivalent values of the risky positions monotone in the risk aversion.