Endogenous fertility endogenous lifetime and econ

Authors: Luciano Fanti Luca Gori

Publish Date: 2013/06/08

Volume: 27, Issue: 2, Pages: 529-564

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Abstract

We examine the effects of child policies on both transitional dynamics and longterm demoeconomic outcomes in an overlappinggenerations neoclassical growth model à la Chakraborty J Econ Theory 1161119–137 2004 extended with endogenous fertility under the assumption of weak altruism towards children The government invests in public health and an individual’s survival probability at the end of youth depends on health expenditure We show that multiple development regimes can exist However poverty or prosperity does not necessarily depend on the initial conditions since they are the result of how a child policy is designed A child tax for example can be used effectively to enable those economies that were entrapped in poverty to prosper There is also a longterm welfaremaximising level of the child tax We show that a child tax can be used to increase capital accumulation escape from poverty and maximise longterm welfare also when a a public payasyougo pension system is in place and b the government issues an amount of public debt Interestingly there also exists a couple child tax–health tax that can be used to find the secondbest optimum optimorum In addition we show that results are robust to the inclusion of decisions regarding the child quantity–quality tradeoff under the assumption of impure altruism In particular there exists a threshold value of the child tax below resp above which child quality spending is unaffordable resp affordable and different scenarios are in existenceThe authors gratefully acknowledge Riccardo Cambini Giam Pietro Cipriani Davide Fiaschi Erasmo Papagni Neri Salvadori and seminar participants at the International Conference in Honour of Salvatore Vinci Poverty Traps An Empirical and Theoretical Assessment and the University of Siena for stimulating discussions and valuable comments on an earlier draft Special thanks go to Mauro Sodini for invaluable comments We acknowledge that this work has been performed within the activity of the PRIN2009 project “Structural Change and Growth” MIUR Ministry of Education Italy The authors are also indebted to two anonymous reviewers and the editor Prof Alessandro Cigno for insightful comments and suggestions on earlier draftsDefine the righthand side of Eq 102 as Gk Then we have 1i G0 = 0 1ii G ″ k 0 for any k 0 1iii lim kto 0+ Gprime k=+infty 1iv lim kto +infty Gprime k=0 and 1v Gprime prime k admits at most three roots and Gprime prime 0ne 0Proposition 1 therefore follows In fact by properties 1i and 1iii zero is always an unstable steady state of Eq 102 By 1ii–1iv Gk is a monotonic increasing function of k and eventually falls below the 45° line so that at least one positive stable steady state exists for any k 0Now assume ad absurdum the existence of an odd number of equilibria By 1ii–1iv there cannot be an odd number of inflection points for any k 0 By property 1v therefore the number of inflection points of Gk is either zero or two for any k 0 Since at least one positive stable steady state exists then for any k 0 the phase map Gk may intersect the 45° line from below at most once before falling below it Hence an even number of equilibria must exist There are either two steady states with the positive one being the unique asymptotically stable equilibrium or at most one positive steady state separates the lowest asymptotically stable steady state from the highest asymptotically stable one and thus the number of equilibria is fourIn addition from Eqs 35 and 36 we observe that if Λ2 0 and Λ3 0 then no inflection points of Gk exist for any k 0 and G ″k 0 and two steady states exist in that case In contrast Gk has two inflection points for any k 0 if at least either Λ2 0 or Λ3 0 is fulfilled and hence in this case four steady states can existIn this appendix we show that the main results of this paper also hold when the government at each date t issues an amount Z t of public debt and levies lump sum taxes tau tz on the young workers Diamond 1965 Jaeger and Kuhle 2009 Spataro and Fanti 2011We have not pursued this analysis further to save space However numerical simulations available on request show that a child tax can be used to permanently escape from poverty and maximise steadystate welfare and their values are higher than when z = 0 Unlike the basic model a child tax now increases the debt per child z / n t due to a lower population growth in the short run However the increase in capital accumulation implies that the ratio of debt per young person over GDP per young person goes down in the long term

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