Solving models with jump discontinuities in policy functions

We show that the Value Function Iteration (VFI) algorithm has difficulties approximating models with jump discontinuities in policy functions. We find that VFI fails to accurately identify both the location and size of jump discontinuities while the Endogenous Grid Method (EGM) and the Finite Element Method (FEM) are much better at approximating this class of models. We further show that combining value function iteration with a local interpolation step (VFI-INT) is sufficient to obtain accurate approximations. Differences between policy functions generated by VFI and these alternative methods are economically significant. We highlight that these differences across methods cannot be identified using Euler equation errors as these are not a sufficient measure of accuracy for models with jump discontinuities in policy functions. As a result, speed comparisons across methods that rely on Euler equation errors as a measure for accuracy can be misleading. The combination of computational speed, relatively easy implementation and adaptability make VFI-INT especially suitable for approximating models with jump discontinuities in policy functions

Securitization and Aggregate Investment Efficiency

This paper studies the welfare properties of competitive equilibria in an economy with incomplete markets subject to idiosyncratic and aggregate shocks. We focus on the role of securitization, whereby borrowers can reduce idiosyncratic asset risk, which enables increased leverage and investment. In the absence of frictions in the securitization process, we show that the ability to securitize assets completes markets. When there are frictions in the market for securitized assets, requiring originators to hold some skin-in-the-game, markets remain incomplete and risk-sharing is limited. In this case, fire-sales are required to repay debt and finance new investments when the economy is hit by a negative shock. Moreover, the equilibrium may be constrained inefficient due to the existence of a pecuniary externality that can result in over or under-investment. We examine policies to correct over-investment and find that a leverage ratio restriction generates a Pareto improvement, while forcing originators to hold additional skin-in-the-game reduces welfare. Both policies reduce leverage and raise prices in a fire-sale, however tightening skin-in-the-game also directly reduces the resources available to those who most need them, which dominates the positive effect of higher prices