Optimal Dynamic Choice of Durable and Perishable Goods

We analyze the life cycle consumption choice model for multiple goods, focusing on the distinction between durables and perishables. As an approximation of the fact that rather high transaction costs and market imperfections prevail in markets for used durables, we assume that investment in durables is irreversible. In contrast to the additive model with one perishable good, the optimal consumption plan is not myopic. Instead, it depends on past as well as on (expected) future prices. The optimal stock level of the durable good is obtained by tracking a certain \emph{shadow level}: The household purchases just enough durables to keep the stock always above this shadow level. It is shown that this shadow level is given by a backward integral equation that replaces the Euler equation. For the perishable good, the `usual' Euler equation determines the optimal choice in terms of the optimal stock of durables. Since the optimal stock level aggregates past as well as future prices, the consumption of perishables ceases to be myopic as well. The solutions show that durables play an important part in intertemporal consumption decisions. In fact, major purchases of durables are being made early in life, whereas no durables are bought in the retirement years. Through substitution and complementarity effects, this has a significant impact on the consumption of perishable goods. On the technical side, the paper provides a new approach to singular control problems that might be widely applicable in other contexts like irreversible investment, price rigidities etc. We present a numerical algorithm that allows one to calculate the shadow level for arbitrary period utility functions and time horizons. Explicit solutions are given for the case of a homogeneous Markov setup with infinite time horizon and Cobb--Douglas type period utilities. This setup includes prices driven by Brownian motion and/or Poisson processes.