In this paper we propose a dynamic partial equilibrium model in order to explain the search behaviour of low-skilled non employed workers, when their employability decreases as the unemployment spell lengthens. We show that the unemployed react to a reduction in their current search effectiveness by decreasing their search efforts, and react to an expected future reduction in search effectiveness by increasing their present search efforts, to exploit the most our of good times. We then separately analyze active and passive labor market policies, consisting in training programs and income support schemes. We show that it is optimal for the government to include among the eligibility criteria for subsidized training a minimum length of the unemployment spell. However, workers should be recruited before they become discouraged and stop searching. We also show that for broad range of the parameters the optimal income support scheme takes the form of unemployment benefits granted for a limited amount of time starting from the beginning of the unemployment spell, coupled with social assistance for long-term unemployed with very limited residual employability.