Bobtcheff, Catherine, Christian Gollier, and Richard Zeckhauser. "Resource Allocation When Projects Have Ranges of Increasing Returns." HKS Faculty Research Working Paper Series RWP08-024, April 2008.

Abstract

A fixed budget must be allocated to a finite number of different projects with uncertain outputs. The expected marginal productivity of capital in a project first increases then decreases with the amount of capital invested. Such behavior is common when output is a probability (of escaping infection, succeeding with an R&D project…). When the total budget is below some threshold, it 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, all projects receive more capital as the budget increases.