A model of on-the-job training in the presence of a minimum wage is presented. This predicts that the minimum wage will have a negative effect on a worker's subsequent wage growth when the labour market is competitive but a U-shaped effect when it is not competitive. This prediction is then tested using data from the Annual Survey of Hours and Earnings in the United Kingdom. Workers who were affected by the minimum wage before age 22 are found to have significantly lower wage growth later in life than others, but only if they worked on jobs that were not covered by a collective labour agreement. Evidence suggests that this difference in wage growth reflects differences in productivity between workers. The results reconcile previous theoretical predictions by Becker and Acemoglu and Pischke.