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 introduce human capital, the knowledge and skills embodied in workers. Chapter 3 adds public capital, the economy’s infrastructure created by the government, such as roads, laws, and utilities.

Appendix

1.1 A—Derivative of the Value Function

Substituting (2.22a, 2.22b) back into U_t gives

$$ {\displaystyle \begin{array}{c}{U}_t=\frac{{\left[{\Psi}_{1t}\left({W}_t-n\left({x}_t+{b}_{t+1}\right)\right)\right]}^{1-1/\sigma }}{1-1/\sigma }+\beta \frac{{\left[{\left(\beta {R}_t\right)}^{\sigma }{\Psi}_{1t}\left({W}_t-n\left({x}_t+{b}_{t+1}\right)\right)\right]}^{1-1/\sigma }}{1-1/\sigma}\\ {}={\Psi}_{1t}^{-1/\sigma}\frac{{\left({W}_t-n\left({x}_t+{b}_{t+1}\right)\right)}^{1-1/\sigma }}{1-1/\sigma }.\end{array}} $$

Next, denote the optimal choices of the intergenerational transfers with an asterisk so that we can write

$$ {\displaystyle \begin{array}{ll}{V}_t\left({W}_t\right)=& {\Psi}_{1t}^{-1/\sigma}\frac{{\left({W}_t-n\left({x}_t^{\ast }+{b}_{t+1}^{\ast}\right)\right)}^{1-1/\sigma }}{1-1/\sigma}\\ {}& +\beta {V}_{t+1}\left({W}_{t+1}H\left({x}_t^{\ast}\right)+\left(1+{r}_{t+1}\right){b}_{t+1}^{\ast}\right).\end{array}} $$

Note that the optimal intergenerational transfers are in general functions of W_t (in the constrained-case, financial transfers are zero and thus do not depend on W_t). However, because the derivative of the right-hand-side with respect to these choices is zero, then we may ignore these indirect effects of W_t when differentiating V_t with respect to W_t (in the constrained-case, the derivative is taken only with respect to human capital investments). Thus, the derivative of the value function with respect to W_t includes only the direct effect, working through U_t in the first expression above. This result is a general one, and is known as the envelope theorem. Equation (2.24) is the envelope theorem applied to V_t+1(W_t+1).

1.2 B—Many-Period Models

In models that extend beyond two periods, it is easier to proceed by thinking of the equilibrium or market clearing condition in terms of goods rather than capital. The two ways of thinking about things are actually equivalent, but the exposition is easier if we conduct the discussion in terms of goods (see Problem 23). This is especially true if one wants to contrast the infinitely-lived agent approach with the overlapping generations’ approach. To reduce notation, we will limit the discussion to the simple model of physical capital accumulation with no human capital or technical progress . In addition, the key points can be made for the case with σ = 1.

As shown in Problem 23, using the goods market approach allows the transition equation for the economy to be written as

$$ {nk}_{t+1}={Ak}_t^{\alpha }-\left(1-\delta \right){k}_t-\frac{C_t}{M_t}. $$

In the overlapping-generations model with two period lifetimes, this equation reduces nicely to a first-order difference equation (Problem 23). This is because the right-hand-side can be reduced to the consumption behavior of a single generation whose consumption depends only on period-t wages and thus only on k_t. The old generation consumes all its income. As a result, their income and their consumption are equal and cancel from the right-hand-side of the transition equation.

However, with more periods of life there will be more generations on the right-hand-side, whose income is less than their consumption, i.e. who save by purchasing capital. The consumption behavior of these generations will depend on wages earned before period-t. For example, think of a model with five periods of life—four working periods and one retirement period. The aggregate consumption behavior on the right-side sums over five different generations. In general, it is only the generation who is retired that will “cancel out.” The saving of other generations will generally not be zero, so all variables that affect their consumption behavior will appear in the transition equation.

Consider the next oldest generation, who is in the last period of its working life. This generation’s working life began three periods ago. Their consumption behavior in period-t will depend on the wages they earned in each of those periods. Thus, wages as far back at period t−3 will appear in the transition equation. The wages in each period are determined by the capital-labor ratio from that period. This implies that the transition equation will be a fifth-order difference equation, including the variables, k_t+1, k_t, k_t−1, k_t−2, k_t−3. The important point is that the state variables, here the capital-labor ratios, characterizing the economy increase as the number of periods of the life-cycle increase. This curse of dimensionality raises the computational complexity of the model when the number of periods in the life-cycle is large.

In contrast, if one assumes that the generations are linked by intergenerational financial transfers, the transition equation of the economy is a second order difference equation , no matter how many periods in the life-cycle are included. To see this, first note that as long as financial intergenerational transfers link the generations together, then we can write C_t = κN_t c_1t (see Problem 21). As periods in the life-cycle are added, only the value of κ changes. For example, κ = 2, with two periods of life, and κ = 5, if there are five periods of life.

Next, solve for c_1t using the goods-market version of the transition equation to get

$$ {c}_{1t}=\frac{M_t}{N_t\kappa}\left[{Ak}_t^{\alpha }-\left(1-\delta \right){k}_t-{nk}_{t+1}\right]. $$

Now substitute into (2.25b) to get

$$ \left[{Ak}_{t+1}^{\alpha }-\left(1-\delta \right){k}_{t+1}-{nk}_{t+2}\right]=\left[\frac{\beta \left(1+\alpha {Ak}_{t+1}^{\alpha -1}-\delta \right)}{n}\right]\left[{Ak}_t^{\alpha }-\left(1-\delta \right){k}_t-{nk}_{t+1}\right], $$

This is a second-order difference equation in k_t that is completely independent of the number of periods in the life-cycle. Thus, the infinitely-lived agent simplification is able to avoid the dreaded curse of dimensionality. A very nifty and useful result.

The lesson is that if you want to do computations, use the infinitely-lived agent approach whenever you can get away with it (e.g. business cycle analysis). Unfortunately, the evidence suggests that for the issues we examine in this book, those that directly focus on private intergenerational transfers and government policies that transfer resources across generations, one must stick with the overlapping generations approach.