Government Debt and Economic Growth in an Overlapping Generations Model.

• Author: Lin, Shuanglin

Shuanglin Lin [*]

Prior studies have shown that an increase in government debt raises the real interest rate and lowers the rate of economic growth. In an overlapping generations model of endogenous growth, this paper shows that an increase in government debt may not increase the real interest rate with the real interest rate being greater than the growth and that an introduction of government debt will increase the growth rate of per capita output if the growth rate is greater than the real interest rate and will decrease the growth rate if the growth rate is less than the real interest rate.

1. Introduction

   Recent decades have seen a large increase in the share of government debt in national income in many countries. This paper develops an endogenous growth model of overlapping generations to examine the effect of government debt on the rate of economic growth. [1]

   Most prior studies of the effects of government policies on economic growth have assumed that government debt is absent. Romer (1986), Jones and Manuelli (1990), King and Rebelo (1990), and Rebelo (1991) showed that when government spending is not productive, an increase in the income tax rate will lower the rate of per capita output growth. [2] Barro (1990) showed that an increase in government spending may either increase or decrease the rate of growth in a model with productive government spending. Government debt was incorporated in endogenous growth models by King (1992) and van der Ploeg and Alogoskoufis (1994). King (1992) developed a model of endogenous growth and showed that a necessary condition for the feasibility of perpetual debt finance is that labor's share of income must be greater than two-thirds. Van der Ploeg and Alogoskoufis (1994) developed a Blanchard-type overlapping generations (OG) model of endogenous growth and found that an increase in government debt, arising from an intertemporal shift in taxation, reduces the growth rate.

   This paper analyzes the effect of government debt by developing a Diamond-type of OG model of endogenous growth. [3] Human capital is the engine of economic growth. At any moment in time, there are three generations: the young generation, the parent generation, and the grandparent generation. All human capital is produced. The parent generation's human capital has a spillover effect on the young generation's human capital accumulation. Government collects taxes and issues debt to finance public expenditures on human capital production. [4] Government debt affects the growth rate by affecting the real interest rate and government spending on human capital production. Unlike the infinitely lived representative agent model, the real interest rate and the growth rate do not necessarily have a positive relationship in the OG model.

   The main results are as follows. When the economy grows endogenously, an increase in government debt does not necessarily increase the real interest rate as shown in exogenous growth models. The relationship between government debt and the growth rate is not monotonic. An introduction of government debt will increase the growth rate, if the initial growth rate is greater than the real interest rate, but will decrease the growth rate if the initial growth rate is less than the real interest rate.

   Section 2 presents the model and defines the steady-state equilibrium. Section 3 provides comparative steady-state analyses of the effects of government debt on the real interest rate and the growth rate of per capita output. Section 4 concludes the paper.

2. The Model

   The economy produces two goods, a physical good, which can be consumed or invested, and human capital (units of effective labor). Individuals live for three periods: accumulating human capital in the first period, working and saving in the second period, and retiring and consuming savings and accrued interest in the third period. Individuals are identical within and across generations, and there is no population growth. The government lives forever, collecting taxes and issuing debt to finance its spending.

   Let [H.sub.t] be the human capital per worker in period t (measured by units of effective labor), and [K.sub.t] be the physical capital per worker in period t. Human capital in period t is owned by generation t - 1 (the parent generation in period t, born in period t - 1), and physical capital in period t is owned by generation t - 2 (the grandparent generation in period t, born in period t - 2). The production function exhibits constant returns to scale in both factors. The output per worker in period t, [Y.sub.t], is:

   [Y.sub.t] = F([H.sub.t], [K.sub.t]) = [H.sub.t]f([k.sub.t]) (1)

   where [k.sup.t] = is [K.sub.t]/[H.sub.t]the ratio of physical capital to human capital. Assume that physical capital is fully depreciated after one production period, as in Sibert (1990) and Lin (1994). Factor markets are perfectly competitive; thus, the rate of returns to each factor is equal to its marginal product; that is,

   1 + [r.sub.t] = [delta][Y.sub.t]/[delta][K.sub.t] = f'([k.sub.t]) (2)

   [w.sub.t] = [delta][Y.sub.t]/[delta][H.sub.t] = f([k.sub.t.]) - [k.sub.t]f'([k.sub.t]), (3)

   where 1 + [r.sub.t] is the rate of return on physical capital in period t and [w.sub.t] is the rate of return on human capital. [5]

   Human capital is determined by four factors, the older generations' human capital, physical capital, government spending, and the young generation's time allocated toward human capital accumulation. [6] The parent generation's human capital affects the young generation's human capital through an external effect. Lucas (1988) emphasized human capital externalities by including human capital externalities in the production of physical goods. The spillover of the older generation's human capital on the young generation's human capital accumulation may occur through many family and social activities. [7]

   As in Glomm and Ravikumar (1992), I assume that each young has one unit of time, part of which is allocated toward leisure and the remaining part toward human capital accumulation. Let [G.sub.t] be the government spending per worker in period t, [l.sub.t] the time allocated for leisure, and 1 - [l.sub.t] the time allocated for human capital accumulation. The output of human capital produced in period t and to be used in period t + 1, [H.sub.t+1], is as follows:

   [H.sub.t+1] = [phi][[(1 - [l.sub.t])[H.sub.t]].sup.[alpha]][[G.sup.1-[alpha]].sub.t] = [phi][(1 - [l.sub.t]).sup.[alpha]][H.sub.t][[g.sup.1-[alpha]].sub.t] (4)

   where 0 [less than] [alpha] [less than] 1 and [g.sub.t] = [G.sub.t]/[H.sub.t], the ratio of government spending to human capital in period t. [8] The human capital of the parent generation, the time spent by the young to accumulate capital and the government spending on human capital production are positively related to the young generation's human capital.

   The economy grows even without population growth. The growth rate of per capita output is defined as: [9]

   [[gamma].sub.t+1] = [Y.sub.t+1] - [Y.sub.t]/[Y.sub.t] = [Y.sub.t+1]/[Y.sub.t] - 1.

   Using Equations 1 and 4, the above equation can be written as:

   [[gamma].sub.t+1] = [H.sub.t+1]/[H.sub.t] f([k.sub.t+1])/f([k.sub.t]) - 1 = [phi][(1 - [l.sub.t]).sup.[alpha]][[g.sup.1-[alpha]].sub.t] f([k.sub.t+1])/f([k.sub.t]) - 1. (5)

   Thus, the growth rate of per capita output depends on the ratio of physical capital to human capital, the ratio of government spending to human capital, the effort of the young (the time allocated toward human capital...