A Simple Model of Liquidity Effects

Summary

We consider here an agent who may invest in a riskless bank account and a share, but may only move money between the two assets at the times of a Poisson process. This models in a simplified way liquidity constraints faced in the real world. The agent is trying to maximise the expected discounted utility of consumption, where the utility is CRRA; this is the objective in the classical Merton problem. Unlike that problem, there is no closed-form solution for the situation we analyse, but certain qualitative features of the solution can be established; the agent should consume at a rate which is the product of wealth and some function of the proportion of wealth in the risky asset, and at the times of the Poisson process the agent should readjust his portfolio so as to leave a fixed proportion of wealth in the risky asset. We establish an asymptotic expansion of the solution in two slightly different formulations of the problem, which allows us to deduce that the ‘cost of liquidity’ is (to first order) inversely proportional to the intensity of the Poisson process.