Equity-efficiency trade-off in quasi-linear environments

I study a simple equity-efficiency problem:  A designer allocates a fixed amount of money to a population of agents differing in privately-observed marginal values for money. She can only screen agents by asking them to burn utility (through some socially wasteful activity). I show that giving a lump-sum payment is outperformed by a mechanism with utility burning when the agent with the lowest money-denominated cost of engaging in the wasteful activity has an expected value for money that exceeds the average value by more than a factor of two.