Fig. 4.

Efficient annuities when saving is unobservable or non-contractible. This figure plots the pair of annuity contracts solving Eq. (5) when savings is unobservable or non-contractible and when V̄^H (high-risk types’ minimum utility) is equal to the utility high-risk types get from their full insurance actuarially fair contract. The annuity curves plot the size of the life-contingent annuity payments as a function of the age of annuitant. “Optimal consumption of ‘deviating’ high-risk types” refers to the optimal consumption pattern for high-risk types who “deviate” and purchase the low-risk types’ equilibrium annuity. Calculations are for the baseline parameters described in Section 4 and Table 1.