Effects of a declining population in a model of economic growth with endogenous human capital - Lucas (1988)

Part of the Contributions to Economics book series (CE)


Human Capital Capital Stock Population Growth Rate Physical Capital Hamiltonian Function 
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  1. 1.
    See Chiang (1992) for a detailed description of problems of dynamic optimization. An introduction to dynamic optimization in continuous time and how it can be used in economic growth theory is given by Barro and Sala-i-Martin (2004), 604f. The behaviour of the household which maximizes its utility over time is based on Ramsey (1928).Google Scholar
  2. 2.
    See Chiang (1992), 12.Google Scholar
  3. 3.
    See Chiang (1992), 261, Solow (2000), 129.Google Scholar
  4. 4.
    In infinite horizon problems the transversality condition needed to provide a boundary condition is typically replaced by the assumption that the optimal solution approaches a steady state; see Kamien and Schwartz (1991), 174.Google Scholar
  5. 5.
    Chiang (1992), 19.Google Scholar
  6. 6.
    See Chiang (1992), 167.Google Scholar
  7. 7.
    See Barro and Sala-i-Martin (2004), 615.Google Scholar
  8. 8.
    See Michel (1982).Google Scholar
  9. 9.
    Barro and Sala-Martin (2003), 617.Google Scholar
  10. 10.
    In the literature, the model is called “Uzawa-Lucas” model as Lucas (1988) is based on Uzawa (1965). Lucas (1988) includes several models. In this thesis, Lucas (1988) refers to chapter 4 of his paper.Google Scholar
  11. 11.
    The production of human capital involves no physical capital. Rebelo (1991) applies a Cobb-Douglas function which employs both human and physical capital in the production function.Google Scholar
  12. 12.
    Lucas (1988), 19.Google Scholar
  13. 13.
    If the accumulation of human capital would follow a rule such as \( \dot h = E\left( {1 - u} \right)h^\zeta \) and ζ < 1 there are diminishing returns to human capital accumulation. This means that the growth rate of human capital is \( \frac{{\dot h}} {h} \leqslant E\left( 1 \right)h^{\zeta - 1} \) so that \( \frac{{\dot h}} {h} \) goes to zero as h grows (see Lucas (1988), 18). See Solow’s criticism of (4.12) in Solow (2000), 126.Google Scholar
  14. 14.
    Lucas (1988), 17. See criticism by Nerlove and Raut (1997), 1140, “... so 10 men who can read are better than 100 who cannot no matter what the size of the labour force?”Google Scholar
  15. 15.
    In Lucas (1988) the production function is Y = A · Kα(uhL)1−αhαγ with technology level A. The term hαγ captures the external effects of human capital. As we are not concerned with externalities of human capital, we can simplify the function to (4.15), i.e. we set γ = 0. In addition we neglect the technology A as it is exogenous in this model.Google Scholar
  16. 16.
    Chiang (1992), 255.Google Scholar
  17. 17.
    Blanchard and Fischer (1989), 38.Google Scholar
  18. 18.
    For example, Barro and Sala-i-Martin (1995), chapter 5.Google Scholar
  19. 19.
    Lucas (1988), 17.Google Scholar
  20. 20.
    Solow (2000), 129.Google Scholar
  21. 21.
    This result is also derived by Robertson (2002) who analyses demographic shocks in the Lucas (1988) model, augmented by unskilled labour.Google Scholar
  22. 22.
    Chiang (1992), 208.Google Scholar
  23. 24.
    Solow (2000, 133) modifies the Lucas (1988) model by introducing leisure time. Then the Lucas model is reduced to the standard Solow model. See also Hahn (1990). See Rebelo (1991) for a model where time is divided between leisure, work and education.Google Scholar

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