Second-, third-, and higher-order consumption functions: a precautionary tale

No analytic solution exists to the consumption problem of an agent who faces uninsurable income shocks, though perturbation methods can be used to derive approximate solutions. This paper shows it is straightforward to extend the well-known second-order consumption function to third order. However, for every n there is a threshold interest rate below which the nth-order correction diverges. This puts a bound on the accuracy that can be achieved with perturbation methods. Even a second-order consumption function can perform spectacularly worse than a zeroth-order function that disregards precautionary saving, and this cannot be rectified by going to higher orders.