We examine the problem of optimally allocating a fixed budget to a finite number of different investment projects whose returns are uncertain and depend on the budget allocated to each of them. The marginal productivity of capital in a project is first increasing then decreasing with the amount of capital invested in it. Such a shape is particularly prevalent when the output is a probability such as the chance of escaping infection or succeeding with a R&D project. When the total budget is below some lower cutoff value, the entire budget is invested in a single project. Above this cutoff, the share invested in a project can be discontinuous and non-monotone in the total budget. Above an upper cutoff budget, all projects receive more capital as the budget increases. If the projects are identical, each will get the same budget. Moreover, under some conditions, the aggregate benefit function also presents first increasing and then decreasing returns to scale as the amount invested increases.