Fig. 2.

A constrained efficient annuity pair. This figure illustrates a constrained efficient pair of contracts (A^L, A^H) solving Eq. (1). A^H lies above A^H*, the full insurance actuarially fair contract for high-risk types. Hence, (A^L, A^H) involves redistribution from low to high-risk types. Since the low-risk (high-risk) types’ indirect utility V^L is lower (higher) with A^L (A^H) than with A^L* (A^{H *}), low-risk (high-risk) types are worse (better) off with (A^L, A^H) than they are with the pair of individually fair full insurance contracts (A^L*, A^H*). When type is observable, (A^L, A^H) is also inefficient (in the sense of Eq. (14)) because each type could be made equally well off with the contract pair (A′^L, A^H) at a lower expected cost to firms; the distance between A′^L and Ā^L (the full insurance contract with the same actuarial cost for the low-risk type as A^L) is a measure of this inefficiency.