Minimum integration income is a major element of the social security system and plays a key role in the French redistributive system. Nevertheless, its potentially discouraging effects on its recipients' working patterns have triggered a debate between those who favour labour market integration in the name of an efficiency or social inclusion principle and those who defend everyone's right to a minimum income in the name of a principle of equity, even at the risk of reducing the level of employment. Some hypotheses suggest that the social optimum justifies the principle of a minimum guaranteed income for individuals whose labour productivity is very low if not zero. This income is presented as a combination of lump sum transfers, means-based transfers and/or a possibly «negative» income tax. We infer labour productivity from observed earned income to try and empirically determine the optimal redistribution for low incomes based on two social welfare functions. The first justifies the current minimum integration income mechanism. The second is found to be favourable to low incomes while giving grounds for the employment premium. Therefore, the minimum income to be guaranteed to the individuals with the lowest productivity and the way in which this income changes in keeping with working patterns are first and foremost dependent on society's redistributive will, i.e. its more or less great aversion to inequality, and on the inequality of incomes before redistribution. The expansion of the initial theoretical framework to encompass a more dynamic perspective shows that the hypothesis of a relatively inert labour supply among the most disadvantaged can be discarded when training is included. Hence, it is no longer work as such that should be encouraged, but also training. A real requalification effort would increase their employability and give them more of an incentive to join the labour market.