Kiyotaki, Michaelides and Nikolov (2011) provide a very simple way of modelling the lifecycle which could be useful for my third-year paper. In KMN’s model households can have either low, medium or high productivity. Each low productivity household switches to medium productivity with probability \(\delta^l\). Medium productivity households switch to high productivity with probability \(\delta^m\). Once a household has switched to high productivity it remains there until retirement. Thus, the transition matrix looks like the one below, which is kinda cool!

\[ \begin{matrix} L: \\ M: \\ H: \end{matrix} \begin{bmatrix} 1-\delta^l & \delta^l & 0 \\ 0 & 1-\delta^m & \delta^m \\ 0 & 0 & 1 \end{bmatrix} \]

All workers have a constant probability of retiring next period, \(1-\omega\). Retirees have a constant probability of dying \(1-\sigma\). Every period there is a flow of new households born with low productivity and no inheritance. KMN’s model has growth, so they allow the number of new borns to exceed the number of households that die, but I probably don’t need to do this. I’m not even sure if I need retirement.

One problem with KMN’s approach, where agent’s face a constant probability of death after retiring, is how to incorporate mortgages that amortize over an agent’s remaining lifetime.