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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.


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.