Task difficulty, performance measure characteristics, and the trade-off between insurance and well-allocated effort

When designing incentives for a manager, the trade-off between insurance and a good allocation of effort across various tasks is often identified with a trade-off between the responsiveness (sensitivity, precision, signal-noise ratio) of the performance measure and its similarity (congruity, congruence) to the benefit of the manager's employer. A necessary condition for the trade-off between responsiveness and similarity to be meaningful is that a perfectly congruent measure creates a higher benefit than an equally responsive non-congruent measure. We show that this condition is met if and only if all tasks are exactly equally difficult and there are no spill-overs or synergies across tasks. This means that for most practical purposes, notions of responsiveness and similarity are not informative about the tradeoff between insurance and allocation. In order to understand this trade-off, task difficulty has also to be taken into account.