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Endogenous lifetime, accidental bequests and economic growth

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Abstract

This paper introduces the concept of unintentional bequests in a closed economy à la Chakraborty (J Econ Theory 116:119–137, 2004) with overlapping generations. We show that scarce public investments in health can lead to poverty traps depending on the relative size of the output elasticity of capital. More importantly, the existence of unintentional bequests, rather than a market for annuities, means that health tax rates play a prominent role in determining the stability of the long-term equilibrium in rich economies. In fact, Neimark–Sacker bifurcations and endogenous fluctuations occur depending on the size of the public health system.

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Notes

  1. As an example, think of the provision of public pensions, which are mainly organised on a pay-as-you-go basis in several European countries: i.e. the income of current workers is taxed by the government to finance the benefits received by the current pensioners (see Fanti and Gori 2012a). There are extensive debates between economists to find appropriate ways to reform the social security system (e.g. Boeri et al. 2001, 2002; Cigno 2007; Cigno and Werding 2007) because of concerns regarding population ageing.

  2. Other major bequest motives are altruism and exchange. While there is no consensus on which motive dominates (see, e.g. Altonji et al. 1997), Hurd (1997) argues that bequests are largely accidental.

  3. Note that, unlike Chakraborty (2004), our model is developed by assuming unintentional bequests without a market for annuities.

  4. This can indeed be assumed in a context of exogenous fertility. Things would be different if fertility was endogenous.

  5. See, e.g. Abel (1985); Chakraborty (2004); Chakraborty and Das (2005); Pestieau et al. (2008); Chakraborty et al. (2010); Fanti and Gori (2012b) for similar formulations of expected utility functions.

  6. Every annuitant deposits his/her savings with a mutual fund. Savings are then invested by the fund to get a return factor (independent of longevity). Then: (i) if an annuitant is alive, he/she gets savings plus the return factor divided by the longevity rate; (ii) if an annuitant dies at the onset of old age, the contract with the fund ends and his/her savings are distributed between all the survived annuitants. The situation is different with accidental bequests, as savings of deceased are directly bequeathed to his/her descendants.

  7. See Michel and de la Croix (2000) and de la Croix and Michel (2002) for a discussion about differences in dynamic outcomes under myopic foresight and perfect foresight in OLG growth models with capital accumulation and two-period lived individuals.

  8. This is shown by several numerical experiments not reported in the paper.

  9. If no eigenvalues of the linearised system around the fixed points of a first order discrete system lie on the unit circle, then such points are defined as being hyperbolic. Roughly speaking, at nonhyperbolic points topological features are not structurally stable.

  10. Since the map is difficult to handle in a neat analytical form, the local stability analysis of the positive (largest) steady state is performed through computations (no closed-form expression of the fixed point exists). With regard to local and global analyses, given that the dynamic patterns are characterised by a possible rich set of complex scenarios, our aim is to describe some interesting outcomes regarding dynamics with no claim of generalisation.

  11. Numerical experiments are of course available upon request.

  12. See Gollin (2002) for estimates on the output elasticity of capital in developed countries.

  13. When \(1/\left({1+\delta }\right)<\alpha <1\) and \(\delta >1\) both the low and high steady states in the model by Chakraborty (2004) are locally asymptotically stable with monotonic trajectories.

  14. For a map G defined on \(U\subset \mathfrak R ^{n}\), a subset \(S\subset U\) is said to be invariant if \(G^{n}(S)\subset S\) for any \(n\in Z\). This invariant set can be a closed (i.e. containing all its accumulation points) curve and like any other invariant set can be locally stable. We refer the reader to Lorenz (1993) or Medio (1995), amongst others, for more formal and rigorous definitions.

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Acknowledgments

The authors are indebted to Michael Kopel, Mauro Sodini, Piero Manfredi, Luigi Bonatti and participants at both the NED11 (Nonlinear Economic Dynamics) conference, held on 1 to 3 June 2011, at the Universidad Politécnica de Cartagena, Spain, and SIE 2011 (Società Italiana degli Economisti), held on 14–15 October, 2011, for stimulating discussions and valuable comments and suggestions on an earlier draft. The authors also acknowledge two anonymous reviewers for insightful comments. The usual disclaimer applies.

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Authors and Affiliations

  1. Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi, 10, 56124 , Pisa, PI, Italy

    Luciano Fanti

  2. Department of Law, University of Genoa, Via Balbi, 30/19, 16126 , Genoa, GE, Italy

    Luca Gori

  3. Department of Economics and Management, University of Pavia, Via San Felice, 5, 27100 , Pavia, PV, Italy

    Fabio Tramontana

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  1. Luciano Fanti
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Correspondence to Luca Gori.

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Fanti, L., Gori, L. & Tramontana, F. Endogenous lifetime, accidental bequests and economic growth. Decisions Econ Finan 37, 81–98 (2014). https://doi.org/10.1007/s10203-012-0138-2

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