For decades, migration economics has stressed the effects of migration restrictions on income distribution in the host country. Recently the literature has taken a new direction by estimating the costs of migration restrictions to global economic efficiency. In contrast, a new strand of research posits that migration restrictions could be not only desirably redistributive, but in fact globally efficient. This is the new economic case for migration restrictions. The case rests on the possibility that without tight restrictions on migration, migrants from poor countries could transmit low productivity ("A" or Total Factor Productivity) to rich countries – offsetting efficiency gains from the spatial reallocation of labor from low to high-productivity places. We provide a novel assessment, proposing a simple model of dynamically efficient migration under productivity transmission and calibrating it with new macro and micro data. In this model, the case for efficiency-enhancing migration barriers rests on three parameters: transmission, the degree to which origin-country total factor productivity is embodied in migrants; assimilation, the degree to which migrants' productivity determinants become like natives' over time in the host country; and congestion, the degree to which transmission and assimilation change at higher migrant stocks. On current evidence about the magnitudes of these parameters, dynamically efficient policy would not imply open borders but would imply relaxations on current restrictions. That is, the new efficiency case for some migration restrictions is empirically a case against the stringency of current restrictions.