Population Ageing and Economic Growth pp 67-87 | Cite as

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

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## Keywords

Human Capital Capital Stock Population Growth Rate Physical Capital Hamiltonian Function
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## References

- 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.See Chiang (1992), 12.Google Scholar
- 3.See Chiang (1992), 261, Solow (2000), 129.Google Scholar
- 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.Chiang (1992), 19.Google Scholar
- 6.See Chiang (1992), 167.Google Scholar
- 7.See Barro and Sala-i-Martin (2004), 615.Google Scholar
- 8.See Michel (1982).Google Scholar
- 9.Barro and Sala-Martin (2003), 617.Google Scholar
- 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.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.Lucas (1988), 19.Google Scholar
- 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.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.In Lucas (1988) the production function is
*Y*=*A*·*K*^{α}(*uhL*)^{1−α}*h*_{α}^{γ}with technology level*A*. The term*h*_{α}^{γ}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.Chiang (1992), 255.Google Scholar
- 17.Blanchard and Fischer (1989), 38.Google Scholar
- 18.For example, Barro and Sala-i-Martin (1995), chapter 5.Google Scholar
- 19.Lucas (1988), 17.Google Scholar
- 20.Solow (2000), 129.Google Scholar
- 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.Chiang (1992), 208.Google Scholar
- 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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