We propose an assignment model in which positions along a hierarchy are attributed to individuals depending on their characteristics. Our theoretical framework can be used to study differences in assignment and outcomes across groups and we show how it can motivate decomposition and counterfactual exercises. It constitutes an alternative to more descriptive methods such as Oaxaca decompositions and quantile counterfactual approaches. In an application, we study gender disparities in the public and private sectors with a French exhaustive administrative dataset. Whereas females are believed to be treated more fairly in the public sector, we find that the gender gap in propensity to get job positions along the wage distribution is rather similar in the two sectors. The gender wage gap in the public sector is 13% and it increases by only 0.7 percentage points when workers are assigned to job positions according to the rules of the private sector. Nevertheless, the gender gap at the last decile in the public sector increases by as much as 3.6 percentage points when using the assignment rules of the private sector.