A Complete One-Sector Neoclassical Growth Model

Abstract

In this chapter we put all the elements discussed in previous chapters together. The model includes private capital from Chap. 2, government capital and taxation from Chap. 3, and fertility and schooling from Chap. 4. The features are combined to study large income differences across rich and poor countries, what is known as development economics. At the dawn of the Industrial Revolution, the differences in per capita income across countries were relatively modest. Per capita income in the richest countries was only 2–4 times greater than per capita income in the poorest countries. Over the course of the last two centuries, per capita income of the rich countries has diverged from that of the poor countries. By the end of the twentieth century, rich countries were 20–40 times richer than poor countries. What could explain stylized growth fact G4, dramatically large gaps in living standards?

Appendix

1.1 Alternative Interpretations of the Cost of Children

In the household model of fertility the cost of children can be thought of as lost adult consumption needed to feed children or forgone work time needed to raise children. In reality, both costs are important but it is simpler in the model to assume just one type of cost. In this chapter we think of the cost of children as representing lost adult consumption that is proportional to parent’s wages because, especially in developing countries, we think of this as the most important cost quantitatively. We use the cost of time rearing in the second half of the book because it gives rise to simpler expressions in the theory and we are not as concerned about quantitative accuracy as we are here.

The key difference between the two cost interpretation surfaces when you calculate the economy’s supply of labor. If the cost of children is lost adult consumption then the family supply of effective labor is

$$ {\widehat{h}}_t\equiv {h}_t+{n}_{t+1}\gamma \overline{h}\left(T-{e}_t\right).\kern3em \left(\mathrm{lost}\ \mathrm{consumption}\ \mathrm{interpretation}\right) $$

Alternatively, under the forgone work time interpretation we have

$$ {\widehat{h}}_t\equiv {h}_t\left(1-\eta {n}_{t+1}\right)+{n}_{t+1}\gamma \overline{h}\left(T-{e}_t\right).\kern3em \left(\mathrm{forgone}\ \mathrm{work}\ \mathrm{time}\right) $$

Note that the labor supply of the adult worker is reduced by the time spent raising children. Using (5.1a), we can simplify the expression for human capital per family as.

$$ {\widehat{h}}_t={h}_t\frac{1+\beta }{1+\beta +\psi }.\kern3.25em \left(\mathrm{forgone}\ \mathrm{work}\ \mathrm{time}\right) $$

1.2 Optimal Fiscal Policy in a Closed Economy

Domestic fiscal policy is determined by maximizing (5.7) 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.

$$ {U}_t={U}_0+{\overline{U}}_t+\left(1+\beta \right)\ln \left(\left(1-{\tau}_t\right){w}_t{D}_t\right)+\beta \ln {R}_t+\psi \ln \left(\left(1-{\tau}_{t+1}\right){w}_{t+1}{D}_{t+1}\right), $$

where is a constant and 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,

$$ \sum \limits_{t=0}^{\infty }{\beta}^t\left(\ln {c}_t^g+\phi \left\{\left(1+\beta \right)\ln \left(\left(1-{\tau}_t\right){w}_t{D}_t\right)+\beta \ln {R}_t+\psi \ln \left(\left(1-{\tau}_{t+1}\right){w}_{t+1}{D}_{t+1}\right)\right\}\right) $$

(5.7′)

subject to (5.4), (5.6), (5.8), and (5.9).

Substituting the constraints into the objective function and collecting common terms yields the following equivalent problem

$$ {\displaystyle \begin{array}{l}\underset{{\left\{{\tau}_{t+1},{g}_{t+1},{k}_{t+1}\right\}}_{t=0}^{\infty }}{\max}\sum \limits_{t=0}^{\infty }{\beta}^t\ln \left[{\tau}_t{k}_t^{\alpha }{g}_t^{\mu \left(1-\alpha \right)}{\widehat{h}}_t-{g}_{t+1}\left(1+q\right){n}_{t+1}\right]\\ {}+\phi \sum \limits_{t=0}^{\infty }{\beta}^t\Big\{\left[\beta \left(\alpha -1\right)+\psi \alpha +\beta \alpha \left(1+\beta \right)\right]\ln {k}_{t+1}\\ {}+\mu \left(1-\alpha \right)\left[\left(\beta +\psi \right)+\beta \left(1+\beta \right)\right]\ln {g}_{t+1}+\left[\beta +\psi +\beta \left(1+\beta \right)\right]\ln \left(1-{\tau}_{t+1}\right)\Big\}\\ {}+\sum \limits_{t=1}^{\infty }{\lambda}_t\left\{\left[\frac{\beta }{1+\beta +\psi}\right]\frac{\left(1-{\tau}_{t-1}\right)\left(1-\alpha \right){k}_{t-1}^{\alpha }{g}_{t-1}^{\mu \left(1-\alpha \right)}{h}_{t-1}}{\left(1+q\right){n}_t{\widehat{h}}_t}-{k}_t\right\},\end{array}} $$

where λ 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 , for t ≥ 1. Be careful to differentiate wherever the choice variable appears in the objective function. Next, substitute into the first order conditions the “guess” , where B is an undetermined coefficient. Finally, solve the first order conditions for B, , , and to get (5.10).

A tricky part of the solution given by (5.10) involves the first order condition for . This equation, along with the first order condition for and the guess for the , can be used to solve for the expression by solving the following difference equation, ${\lambda}_t{k}_t={\beta}^{t-1}\left\{\frac{\alpha \beta}{1-B}+\phi \left[\beta \left(\alpha -1\right)+\alpha \psi +\alpha \beta \left(1+\beta \right)\right]\right\}+{\alpha \lambda}_{t+1}{k}_{t+1}$, to get ${\lambda}_t{k}_t=\frac{\beta^{t-1}}{1-\alpha \beta}\left\{\frac{\alpha \beta}{1-B}+\phi \left[\beta \left(\alpha -1\right)+\alpha \psi +\alpha \beta \left(1+\beta \right)\right]\right\}$. Use this solution to eliminate in the first order conditions for and . Using the guess for , these two first order conditions can be used to solve for and B to get ${\tau}_t=\tau =\frac{1-\alpha \beta}{1+\left(1-\alpha \beta \right)\left(1-B\right)\phi \Gamma}$ and $\mathrm{B}=\frac{\frac{\mu \beta \left(1-\alpha \right)}{1-\alpha \beta}+\mu \beta \left(1-\alpha \right)\phi \Gamma}{1+\mu \beta \left(1-\alpha \right)\phi \Gamma}$. Combining these two expressions, completes the solution.

1.3 Optimal Fiscal Policy in an Open Economy

In an open economy , the government’s problem can be written so that it solves

$$ {\displaystyle \begin{array}{l}\underset{{\left\{{\tau}_{t+1},{g}_{t+1}\right\}}_{t=0}^{\infty }}{\max}\sum \limits_{t=0}^{\infty }{\beta}^t\ln \left[{\tau}_t{\left(\frac{\left(1-{\tau}_t\right)\alpha }{r\ast}\right)}^{\frac{\alpha }{1-\alpha }}{g}_t^{\mu }{\widehat{h}}_t-{g}_{t+1}\left(1+q\right){n}_{t+1}\right]\\ {}+\phi \left[\psi +\beta \left(1+\beta \right)\right]\sum \limits_{t=0}^{\infty }{\beta}^t\left\{\frac{1}{1-\alpha}\ln \left(1-{\tau}_{t+1}\right)+\mu \ln {g}_{t+1}\right\}.\end{array}} $$

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 and generates first order conditions. As before guess a solution for g of the form.

$$ \left(1+q\right){n}_{t+1}{g}_{t+1}=B{\tau}_t{\left(\frac{\alpha \left(1-{\tau}_t\right)}{r\ast}\right)}^{\frac{\alpha }{1-\alpha }}{g}_t^{\mu }{\widehat{h}}_t. $$

Substitute into the first order conditions and solve for and B to get the solution in the text.