We consider an economy in which the technology exhibits nonconvexities due to fixed costs associated with production. Taking into account the incentives for investment to decrease fixed costs, we characterize the circumstances under which an underdeveloped economy can catch up with the developing ones. We show that it is optimal to get rid of the fixed costs inherent in production in finite time provided that the initial level of fixed costs are not too high and the technology for reducing fixed costs is sufficiently efficient. Indeed, we obtain that even though the income disparities may be very persistent and can be perceived as poverty traps, economies with not very high initial fixed costs and sufficiently efficient technology for reducing fixed costs would ultimately converge to the same steady state level of per capita income.