Before revision:  The decision rule is based on thresholding the sum of transformed valuations of the agents. We show that the rate of change of the threshold is the key to characterizing the trade-off between balancing the budget and maximizing the welfare .
After revision:  The decision rule is based on thresholding the sum of transformed valuations exceeds a threshold that only depends on the population size n. The increasing function can be chosen arbitrarily. For example, it can simply be the identity function. Our results show that the rate of change of the threshold is the key to characterizing the trade-off between balancing the budget and maximizing the welfare .