A Certified Howard Algorithm for Epstein–Zin Preferences

This paper combines the variational utility representation of Geoffard (1996) and Dumas et al. (2000) with a variation of the Howard algorithm given in Bokanowski et al. (2009) to obtain a globally convergent algorithm for finite-dimensional Bellman equations arising from continuous-time problems with Epstein–Zin preferences. The variational representation of recursive utility introduces an auxiliary discount rate that makes policy evaluation linear. However, feasible discount policies can generate evaluation matrices that fail to be monotone. Our certification argument uses current value iterates or barrier functions to establish monotonicity of the matrices generated by the algorithm and provides conditions under which the value iterates converge monotonically to the unique admissible solution. We then illustrate the method in a consumption-savings problem with undiversifiable income risk and derive an explicit coefficient for the square-root behavior of savings near the borrowing constraint for agents with the lowest income. An extension to permanent income risk shows that the choice of initialization matters: the certified initial discount rates form an explicit interval that excludes the CRRA discount rate once permanent risk is large.