Research suggests that, at the levels set in countries like the US and the UK, minimum wages have little effect on employment but do have impacts on wage inequality. However we lack models that can explain these facts – this paper presents one based on imperfect labour markets. The paper also investigates the impact of the UK’s National Minimum Wage on wage inequality finding it can explain a sizeable part of the evolution of wage inequality in the bottom half of the distribution in the period 1998-2010. We also present evidence that the impact of the NMW reaches up to 40% above the NMW in 2010 which corresponds to the 25th percentile. These spillovers are larger in low-wage segments