Abstract
In this chapter, we extend the basic neoclassical growth model in several ways. In the previous chapter, we learned that human capital is an important source of economic growth. Here, we introduce a simple theory of human capital formation based on parents desire to invest in the “quality” or economic productivity of their children. Parents will also choose the “quantity” of their children, giving us a theory of fertility that makes population growth, another important determinant of economic growth identified in Chap. 2, endogenous. Recall that high population growth makes it difficult to accumulated capital per worker and thus slows the growth in per capita income.
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- 1.
Galor and Moav (2002) generalize this specification by allowing for a separate utility weight on the quantity and quality of children. They then go on to develop an evolutionary theory in which households raise the weight they placed on the quality of their children over the course of economic development. Using this more flexible specification would increase the ability of our model to fit the stylized growth facts.
- 2.
For notational simplicity only, we assume the government’s time discount factor is the same as that used by private households. One could allow the discount factor to differ from private households to study how the government’s time preference affects policy.
- 3.
We do not study optimal government debt or monetary policy in this setting. These polices are obviously important extensions left for future work.
- 4.
We assume that the government can commit to its policy choices in advance. For a discussion of commitment issues in regard to the setting of fiscal policy see Lundquist and Sargent (2004, Chap. 22).
- 5.
There is also the issue of differences in the tax base across rich and poor countries. Poor countries have much larger informal sectors that go untaxed. This causes poor countries to collect a small fraction of output in taxes with the same tax rate. An informal sector is needed to capture this and other important features of poor economies. We discuss this extension in the conclusion.
- 6.
The large effects on worker productivities are accentuated in a three-period model. With only three periods, the population of households that are saving compared to the future work force is unrealistically small. In addition, the high fertility rate, without a high rate of mortality, will imply a large increase in the size of the future workforce. Both these features cause the capital accumulation financed by the current period’s saving to be more thinly spread over the next generation of workers than in a model with many periods of work (and saving) and with the high death rates that mediate population growth in poor countries.
- 7.
In 1997 Mexico began Progresa, a program designed to increase human capital in poor families by paying families to send their children to school and to visit health care providers. Grants are provided directly by the government to the mothers of children. The school grants cover about 2/3 of what the child would receive in full time work (Krueger 2002).
- 8.
Our model abstracts from tuition costs (see Problem 5). The government can raise schooling by increasing tuition subsidies. Doepke (2004) and Lord and Rangazas (2006) study the historical impact of government tuition subsidies in England. They find that lower tuition has modest effects on schooling and growth. Lower tuition reduces the cost of all children and, in particular, young children who would have attended school in any case. This raises fertility for several periods and slows the demographic transition. Thus, something like a Progresa program or compulsory schooling is needed to generate a quick demographic transition and rapid economic growth.
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Appendix*
Appendix*
1.1 Optimal Fiscal Policy in a Closed Economy
Domestic fiscal policy is determined by maximizing (3.7) subject to the government budget constraint and the accumulation equations for private and public capital. The private household’s indirect utility function may be written as
where \( U_{0} \) is a constant, and \( \bar{U}_{t} = (1 + \beta )\ln h_{t} + \psi \ln n_{t + 1} + \psi \ln h_{t + 1} \) is independent of fiscal policy. For the purpose of setting optimal fiscal policy, the government can then be modeled as choosing tax rates and public capital to maximize,
subject to (3.4a, b), (3.6), (3.8), and (3.9).
Substituting the constraints into the objective function and collecting common terms yield the following equivalent problem
, where \( \lambda \) is the multiplier associated with the private capital accumulation constraint.
To solve this sequence problem, begin by differentiating to get the first-order conditions for \( \tau_{t} ,g_{t} ,k_{t} ,\lambda_{t} \), for \( t \ge 1 \). Be careful to differentiate wherever the choice variable appears in the objective function. Next, substitute into the first-order conditions the “guess” \( (1 + q)n_{t + 1} g_{t + 1} = B\tau_{t} k_{t}^{\alpha } g_{t}^{\mu (1 - \alpha )} \hat{h}_{t} \), where B is an undetermined coefficient. Finally, solve the first-order conditions for B, \( \tau_{t} \), \( g_{t} \), and \( k_{t} \) to get (3.10a–c).
A tricky part of the solution given by (3.10a–c) involves the first-order condition for \( k_{t} \). This equation, along with the first-order condition for \( \lambda_{t} \) and the guess for the \( g_{t} \), can be used to solve for the expression \( \lambda_{t} k_{t} \) by solving the following difference equation, \( \lambda_{t} k_{t} = \beta^{t - 1} \left\{ {\frac{\alpha \beta }{1 - B} + \phi \left[ {\beta (\alpha - 1) + \alpha \psi + \alpha \beta (1 + \beta )} \right]} \right\} + \alpha \lambda_{t + 1} k_{t + 1} \), to get \( \lambda_{t} k_{t} = \frac{{\beta^{t - 1} }}{1 - \alpha \beta }\left\{ {\frac{\alpha \beta }{1 - B} + \phi \left[ {\beta (\alpha - 1) + \alpha \psi + \alpha \beta (1 + \beta )} \right]} \right\} \). Use this solution to eliminate \( \lambda_{t + 1} k_{t + 1} \) in the first-order conditions for \( \tau_{t} \) and \( g_{t} \). Using the guess for \( g_{t} \), these two first-order conditions can be used to solve for \( \tau_{t} \) and B to get \( \tau_{t} = \tau = \frac{1 - \alpha \beta }{{1 + (1 - \alpha \beta )(1 - B)\phi {\varGamma }}} \) and \( {\text{B}} = \frac{{\tfrac{\mu \beta (1 - \alpha )}{1 - \alpha \beta } + \mu \beta (1 - \alpha )\phi {\varGamma }}}{{1 + \mu \beta (1 - \alpha )\phi {\varGamma }}} \). Combining these two expressions completes the solution.
1.2 Optimal Fiscal Policy in an Open Economy
In an open economy, the government’s problem can be written so that it solves
This problem differs from the closed economy problem because private capital intensity is now determined by international capital flows rather than domestic saving. In a closed economy, government policy affected private capital formation by affecting the after-tax wage of savers that fund the subsequent period’s private capital intensity. Now, government policy affects private capital intensity by affecting the marginal product of private investments in the poor country—reduced by higher tax rates and raised by higher public capital intensity. In an open economy , government policy has a more immediate effect on private capital formation—this period’s policy affects this period’s capital intensity rather than this period’s saving flow and next period’s capital intensity.
Differentiating with respect to \( \tau_{t + 1} \) and \( g_{t + 1} \) generates first-order conditions. As before, guess a solution for g of the form
Substitute into the first-order conditions and solve for \( \tau_{t + 1} \) and B to get the solution in the text.
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Das, S., Mourmouras, A., Rangazas, P.C. (2015). Extensions to Neoclassical Growth Theory. In: Economic Growth and Development. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-14265-4_3
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