Immigration provides opportunities to increase knowledge production. But this depends on an increased demand for knowledge and on immigrants expanding the supply of skills not otherwise available. Unfortunately, this is not always the case. The first problem is university-sector demand: If universities do not take advantage of a supply increase by expanding faculty lines, new scientists can crowd out current ones, with little change in knowledge. This happened around 1991 when >1000 Soviet-trained mathematicians emigrated, many interacting with Western scientists for the first time. We learned the effect of this shock by comparing subfields of U.S. mathematics that the Soviets specialized in with those they knew nothing about (1). U.S. institutions eagerly hired Soviets with the best curricula vitae, especially midcareer mathematicians who had already excelled. But without an expansion in faculty slots, the only slots available were not those already taken by inferior tenured mathematicians, but rather, slots that would have been taken by newly minted Ph.D.'s. Because young scientists have more years of productivity ahead of them than do older ones, this proved disappointing: The new knowledge produced by Soviet émigrés was at best on par with the knowledge that would have been produced by the young mathematicians who lost or never got positions or who went to inferior research jobs. Average output of the most-affected American mathematicians declined by about one-third. The second problem is supply in the for-profit sector: If the U.S. H-1B visa program brings individuals with skills already common in the United States, then the potential for the firms that hire them to produce more knowledge than they otherwise would have is limited. We compared firms that randomly received access to H-1B visa immigrants to those that randomly did not (2). The supply of workers with similar skills was sufficiently prevalent that firms that missed out on hiring H-1B immigrants were able to quickly hire someone else. At the firms that hired them, the median H-1B visa employee crowded out approximately 1.5 other employees, with no increase in the firms' patenting or patent citations in subsequent years.