An Analytical Introduction to Dynamic Optimization: The Robinson Crusoe Economy

Abstract

Dynamic optimization is an essential tool of modern macroeconomics. We suggest to teach this technique based on a simplified version of the Uzawa-Lucas model of endogenous growth. Using this model it is possible to present the key concepts of dynamic optimization without applying linearization techniques. Even though the model assumes a concave utility function and a neoclassical production technology, we will state the stable and the unstable solution branches in the phase diagram in closed-form. Saddle path behaviour and the role of transversality conditions can therefore be studied extensively without using complicated mathematical tools. We also show how homogeneity in the initial conditions, a property that applies to almost all macroeconomic models, can be used to introduce a simple numerical solution method.

Keywords: Closed-form solution, saddle path stability, homogeneity in initial conditions

JEL Classification: A23, C61, C63