Extensions to Neoclassical Growth Theory

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.

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

$$ U_{t} = U_{0} + \bar{U}_{t} + (1 + \beta )\ln w_{t} D_{t} + \beta \ln (1 + r_{t + 1} ) + \psi \ln w_{t + 1} D_{t + 1} , $$

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,

$$ \sum\limits_{t = 0}^{\infty } {\beta^{t} \left( {\ln c_{t}^{g} + \phi \left\{ {(1 + \beta )\ln w_{t} D_{t} + \beta \ln (1 + r_{t + 1} ) + \psi \ln w_{t + 1} D_{t + 1} } \right\}} \right)} $$

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

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\left\{ {\tau_{t + 1} ,g_{t + 1} ,k_{t + 1} } \right\}_{t = 0}^{\infty } }} & \sum\limits_{t = 0}^{\infty } {\beta^{t} \ln \left[ {\tau_{t} k_{t}^{\alpha } g_{t}^{\mu (1 - \alpha )} \hat{h}_{t} - g_{t + 1} (1 + q)n_{t + 1} } \right]} \\ & \quad + \phi \sum\limits_{t = 0}^{\infty } {\beta^{t} } \{ \left[ {\beta (\alpha - 1) + \psi \alpha + \beta \alpha (1 + \beta )} \right]\ln k_{t + 1} + \mu (1 - \alpha )\left[ {(\beta + \psi ) + \beta (1 + \beta )} \right]\ln g_{t + 1} \\ & \quad + \left[ {\beta + \psi + \beta (1 + \beta )} \right]\ln (1 - \tau_{t + 1} )\} \\ & \quad + \sum\limits_{t = 1}^{\infty } {\lambda_{t} } \left\{ {\left[ {\frac{\beta }{1 + \beta + \psi }} \right]\frac{{(1 - \tau_{t - 1} )(1 - \alpha )k_{t - 1}^{\alpha } g_{t - 1}^{\mu (1 - \alpha )} h_{t - 1} }}{{(1 + q)n_{t} \hat{h}_{t} }} - k_{t} } \right\} \\ \end{aligned} $$

, 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

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{\left\{ {\tau_{t + 1} ,g_{t + 1} } \right\}_{t = 0}^{\infty } }} & \sum\limits_{t = 0}^{\infty } {\beta^{t} \ln \left[ {\tau_{t} \left( {\frac{{(1 - \tau_{t} )\alpha }}{{1 + r^{w} }}} \right)^{{\tfrac{\alpha }{1 - \alpha }}} g_{t}^{\mu } \hat{h}_{t} - g_{t + 1} (1 + q)n_{t + 1} } \right]} \\ & \quad + \phi \left[ {\psi + \beta (1 + \beta )} \right]\sum\limits_{t = 0}^{\infty } {\beta^{t} } \left\{ {\tfrac{1}{1 - \alpha }\ln (1 - \tau_{t + 1} ) + \mu \ln g_{t + 1} } \right\} \\ \end{aligned}. $$

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

$$ (1 + q)n_{t + 1} g_{t + 1} = B\tau_{t} \left( {\frac{{\alpha (1 - \tau_{t} )}}{{1 + r^{w} }}} \right)^{{\tfrac{\alpha }{1 - \alpha }}} g_{t}^{\mu } \hat{h}_{t} . $$

Substitute into the first-order conditions and solve for $\tau_{t + 1}$ and B to get the solution in the text.