The non-linear Calvo model at the zero bound: Some analytic solutions and numerical results

This paper shows closed form solutions for the non-linear Calvo model at the zero bond under the assumption that uncertainty is given by a two state Markov chain with an absorbing state. This allows us to explicitly compare the solution of the non-linear model to the better known log-linearized version. In line with the log-linear model, we confirm in the non-linear setting i) large drops in output as shocks become more persistent until bifurcation occurs, ii) large government spending multipliers that have to be above 1 and iii) the paradox of toil. These results are in contrast with some recent literature on non-linearities at the ZLB. Mostly this is because that literature assumes particular form of Rotemberg prices which leads to "implausible" large cost of price adjustment in a way we make precise. Overall the non-linear Calvo model behaves similarly as its linearized counterpart both qualitatively and quantitatively with some important caveats we document.