A skill-biased change in technology can account at once for the changes observed in a number of important variables of the US labour market between 1970 and 1990. These include the increasing inequality in wages, both between and within education groups, and the increase in unemployment at all levels of education. In contrast, in previous literature this type of technology shock cannot account for all of these changes. The paper uses a matching model with a segmented labour market, an imperfect correlation between individual ability and education, and a fixed cost of setting up a job. The endogenous increase in overeducation is key to understand the response of unemployment to the technology shock.