We propose and estimate a model of demand and supply of annuities. To this end, we use rich data from Chile, where annuities are bought and sold in a private market via a two-stage process: first-price auctions followed by bargaining. We model firms with private information about costs and retirees with different mortalities and preferences for bequests and firms' risk ratings. We find substantial costs and preference heterogeneity, and because there are many firms, the market performs well. Counterfactuals show that simplifying the current mechanism with English auctions and "shutting down" risk ratings increase pensions, but only for high-savers.