Spline Approximation of Policy Functions in Economic Dynamics with Uncertainties

Modern economic theory views the economy as a dynamical system in which rational decisions are made in the face of uncertainties. The dynamics includes changes over time of market behavior such as consumption, investment, labor supply, and technology innovation, all interpreted in a broad sense. The Euler equation arises as the first order optimality condition when solving an economic dynamics system. Finding the policy function inherent in the Euler equation is an important but challenging task. This note proposes a Newton iterative scheme on approximating the unknown policy functions by composite 1-dimensional cubic splines. This spline approach has the advantages of freedom in the node collocation, simplicity in the derivative calculation, fast convergence, and high precision over the conventional projection methods. Applications to the neoclassical growth model with leisure choice are used to demonstrate the working of the idea. In particular, tensor products are employed to simplified and effectuate the operations.