Tensor Spline Approximation 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. Optimizing decisions over time on market behavior such as consumption, investment, labor supply, and technology innovation is of practical importance. Interpreting all market behavior in a broad sense, the problem finds further applications in many areas other than economics. Finding the policy function inherent in the associated Euler equation has been an important but challenging task. This note proposes using composite 1-dimensional cubic splines in tensor form to process the Newton iterative scheme on approximating the unknown policy functions. This tensor 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 throughout to simplify and effectuate the operations.