Existence, Uniqueness, and Computational Theory for Time Consistent Equilibria: A Hyperbolic Discounting Example

We present an asymptotically valid analysis of a simple optimal growth model with hyperbolic discounting. We use the implicit function theorem for Banach spaces to show that for small deviations from exponential discounting there is a unique solution near the exponential discounting solution in the Banach space of consumption functions with bounded derivatives. The proof is constructive and produces both an infinite series characterization and a perturbation method for solving these problems. The solution uses only the contraction properties of the exponential discounting case, suggesting that the techniques can be used for a wide variety of time consistency problems. We also compare the computational procedure implied by our asymptotic analysis to previous methods. Finally, we present a simple tax policy example that illustrates how to apply the method more generally.