Overlapping-Generations Model of Economic Growth

Abstract

This chapter introduces the one-sector neoclassical growth model with overlapping generations. The primary focus of the chapter is growth via private physical capital accumulation. We think of private physical capital as manmade durable inputs to the production process. For our purposes, private capital can be primarily thought of as plant and equipment that is produced in one period and then used in production in the following period. To model production, we introduce firms, economic institutions that combine physical capital and labor to produce goods and services. Later in the chapter, we re-introduce the public capital that was the focus of Chaps. 2 and 3 and study the interaction between public and private capital accumulation, along with other effects of fiscal policy on growth.

Appendix

4.1.1 The Government Intertemporal Budget Constraint

To begin construction of the GIBC , write the period t + 1 version of (4.15),

$$ {B}_{t+2}+{\tau}_{t+1}{w}_{t+1}{D}_{t+1}\left(1+\varepsilon \right){N}_{t+1}={R}_t{B}_{t+1}+{z}_{t+1}{N}_t+{w}_{t+1}{D}_{t+1}\varepsilon {N}_{t+1}+{G}_{t+2}. $$

Next solve for B _t + 1 in (4.15) and substitute the solution into the equation above and rearrange terms to get

$$ \frac{B_{t+2}}{R_t{R}_{t-1}}={B}_t+\frac{PD_t}{R_{t-1}}+\frac{PD_{t+1}}{R_t{R}_{t-1}}, $$

where PD _t  ≡ z _t N _t − 1 + w _t D _t εN _t  + G _t + 1 − τ _t w _t D _t (1 + ε)N _t is the primary deficit: the difference between spending, excluding interest and debt repayments, and taxes. We can continue this process of “solving forward” by substituting the expression above into the period t + 2-version of (4.15) and so on. The end result of the forward substitution, N-periods ahead, gives

$$ \frac{B_{t+N}}{\prod \limits_{i=0}^N{R}_{t-1+i}}={B}_t+\sum \limits_{i=0}^{N-1}\frac{PD_{t+i}}{\prod \limits_{j=0}^i{R}_{t-1+j}}. $$

To continue the forward substitution out to the indefinite future, the left-hand-side of the equation above, the present value of outstanding government debt in period t + N, cannot “explode.” In other words, government debt cannot become “too large” in present value terms. This requires that growth rate of debt be smaller than the interest rate, so that

$$ \frac{B_{t+N}}{\prod \limits_{i=0}^N{R}_{t-1+i}}\to 0,\kern1em \mathrm{as}\kern1em N\to \infty . $$

This condition, known as the No Ponzi Game (NPG) condition, means that the government cannot continually issue new debt that is large enough to pay back both previously issued debt and the interest owed on previously issued debt. This scenario would be like the famous Ponzi schemes in finance where funds collected from new investors are used to pay off previous investors. If the government could get away with this much borrowing, it is not constrained at all. Note that satisfying the condition does allow the government to “rollover” a finite amount of debt forever, as long as it finances the interest on that debt with taxes so that the growth rate of the debt is not equal to the interest rate or greater. Using the condition that the present value of government debt goes to zero as time marches on, allows us to write the GIBC as in the text.

There is a related requirement that says for the government to remain solvent, the debt-to-output ratio must remain finite and not explode over time. It is possible for the government to remain solvent even if the NPG condition fails. This could happen if the growth rate of output exceeds the interest rate on government debt . There have been extended historical episodes where the growth rate of output has exceeded the real interest rate on debt. However this is normally not possible so it is also not possible for the government to indefinitely borrow ever larger amounts to pay back both past debt and interest. For example, currently real interest rates are less than the growth in output. As mentioned in this Chapter and as will be discussed in detail in Chap. 7, this situation will not last because there are factors that will both raise interest rates and lower output growth rates. The likelihood of a sharp rise in interest rates increases with the continued increase in debt to GDP ratio.

4.1.2 Tax Rates

In the text, we consider the value of the wage tax rate that maximizes the height of the transition equation for the private capital-labor ratio. Maximizing the growth in private capital intensity is not necessarily a reasonable objective. Instead we might consider the tax rate that maximizes state worker productivity (τ ^∗∗∗) or steady state household utility (τ ^∗∗). One can compute these tax rates as well (see Problems 29–31). The comparison of the three tax rates is

$$ {\displaystyle \begin{array}{l}{\tau}^{\ast \ast \ast }=\frac{\frac{\mu \left(1-\alpha \right)}{\mu \left(1-\alpha \right)+\alpha }+\varepsilon }{1+\varepsilon }>\\ {}{\tau}^{\ast \ast }=\frac{\frac{\mu \left(1-\alpha \right)}{\mu \left(1-\alpha \right)+\alpha}\left(\frac{1+\beta }{\left(\frac{1}{\mu \left(1-\alpha \right)+\alpha}\right)+\beta}\right)+\varepsilon }{1+\varepsilon }=\frac{\mu \left(1-\alpha \right)\left(\frac{1+\beta }{1+\beta \left(\mu \left(1-\alpha \right)+\alpha \right)}\right)+\varepsilon }{1+\varepsilon }>\\ {}{\tau}^{\ast }=\frac{\mu \left(1-\alpha \right)+\varepsilon }{1+\varepsilon },\end{array}} $$

because μ(1 − α) + α < 1.

The tax rate that maximizes steady utility is perhaps the most compelling. It is higher than the tax rate that maximizes steady state capital intensity because there is a benefit to households of keeping the private capital intensity lower than the maximum. All households are savers, so a higher return to capital, other things constant, raises household welfare. The desire to keep the return to capital high creates an incentive to keep private capital intensity low. This consideration causes the policy maker to set the tax rate higher than the one that maximizes the steady state value of k.

The highest tax rate is the one that maximizes steady state worker productivity . This tax rate is higher than the rate that maximizes steady state utility because it does not account for the fact that a higher tax rate on wages lowers the after-tax wage that determines household consumption and instead only focuses on the before-tax wage associated with worker productivity .