Homogeneity, Saddle Path Stability, and Logarithmic Preferences in Economic Models

In a stylized Robinson Crusoe economy, we demonstrate the usefulness of homogeneity in initial conditions when solving and analyzing macroeconomic models. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimization problem. The model’s solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches and the structure of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuoustime Hamilton-Jacobi-Bellman equation. In this case the utility maximization problem can be transformed into an initial value problem for an ordinary differential equation.