When the age of death is uncertain, individuals will leave bequestseven if they have no desired bequestssimply because they will hold wealth against the possibility of living longer. Bequests are accidental. Starting from a baseline level of Social Security benefits, an increase in benefits will cause consumption to increase. However, consumption may not increase by as much as the increase in Social Security, which would cause wealth to be greater than under the baseline scenario. The higher wealth levels would translate into greater bequests. Therefore, an increase in Social Security benefits may not be a complete transfer from the younger generation to the older generation: some of the increase in benefits may be bequeathed back to the younger generation. Whether this happens depends on the form of the utility function, the amount of bequeathable wealth, and whether there is a bequest motive. The objective of this paper is to quantify for single persons how much of an increase in Social Security benefits would be bequeathed back to the younger generation. We find that, at least for singles, increases in Social Security benefits are unlikely to be offset by bequests.