Risk Sharing, Inequality, and Fertility

We use an extended Barro-Becker model of endogenous fertility, in which parents are heterogeneous in their labor productivity, to study the efficient degree of consumption inequality in the long run when parents productivity is private information. We show that a feature of the informationally constrained optimal insurance contract is that there is a stationary distribution over per capita continuation utilities – there is an efficient amount of long run inequality. This contrasts with much of the earlier literature on dynamic contracting where ‘immiseration’ occurs. Further, the model has interesting and novel implications for the policies that can be used to implement the efficient allocation. Two examples of this are: 1) estate taxes are positive and 2) there are positive taxes on family size. We are indebted to Alice Schoonbroodt for all of her helpful comments and suggestions at various stages of this project. We would also like to thank Laurence Ales, V.V. Chari, Bob Lucas, Mike Golosov, Greg Kaplan, Chris Phelan, Richard Rogerson, Aleh Tsyvinski, Ariel Zetlin-Jones and seminar participants at ASU, Colubmia, Carnegie Mellon, Iowa State, Ohio State, Texas Austin, Western Ontario, Wharton, Yale, 2009 SED summer meeting and 2009 Minnesota Macro Workshop for comments.