Theoretical and Computational Appendix to: “Risk Sharing: Private Insurance Markets or Redistributive Taxes?”

In this appendix we provide more extensive proofs of results in the main paper. In particular we explicitly prove results that are direct adaptations of arguments from Atkeson and Lucas (1995) and hence omitted in the main paper We also provide a detailed discussion of the algorithm used in the quantitative exercises of the main papers. 1 Proofs1 1.1 Optimal Policies Induce E¢cient Allocations In this subsection we prove our claim in Section 3.1 of the main text that a stationary allocation fht(w0; y)gt=0 is e¢cient if it is induced by an optimal policy (h; fgy0g) from the functional equation (16) with R > 1 and satis...es the resource constraint with equality. The ...rst step is to ...nd an operational way to solve for e¢cient allocations, which will then lead us to the recursive problem (16): Consider the problem of a social planner faced with a sequence of intertemporal shadow prices fRtgt=0 to minimize the value of resources needed to deliver expected discounted utility ¤We would like to thank Andrew Atkeson for helpful discussions. All errors are our own 1 Equation numbers refer to equations in the main paper, unless they are preceded by an A; in which case they refer to equations in this appendix.