Endogenous Growth in an OLG Model with a Fixed Factor

Abstract

This paper examines the conditions for endogenous growth in an overlapping generations (OLG) model with two sectors of production when the returns to scale may be non constant and provides the technological conditions for the highest in- come growth rate.

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Centre de Recherche en Economie Publique et de la Population






CREPP Working PaPer Series
CREPP WP No 2010/02

Endogenous Growth in an OLG Model with a Fixed Factor

Lionel Artige

August 2010
CREPP - Centre de recherche en Economie Publique et de la Population
Boulevard du Rectorat 7 (B31)
4000 Liège Belgium

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Endogenous Growth in an OLG Model with a Fixed
Factor∗
Lionel Artige
HEC - Universit´ e de Li` ege
August 3, 2010
Abstract
This paper examines the conditions for endogenous growth in an overlapping
generations (OLG) model with two sectors of production when the returns to scale
may be non constant and provides the technological conditions for the highest in-
come growth rate.
Keywords: endogenous growth, increasing returns, human capital, land, over-
lapping generations.
JEL Classification numbers: E13, O11, O41.
∗I am grateful to David de la Croix, Klaus Desmet, Philippe Michel, Philippe Monfort and Henri
Sneessens for helpful comments. Responsibility for all errors is my own. Address for correspondence:
HEC - Universit´ e de Li` ege, Economics department, Boulevard du Rectorat,7, B31, 4000 Li` ege, Belgium.
Email: lionel.artige@ulg.ac.be

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1 Introduction
In two-sector growth models, constant returns to scale technology applying to both sectors
implies long-term growth when the reproducible factors grow at the same rate.1
structure of these models is generally the same. One of the two sectors produces the
physical capital and the other sector produces an investment good that varies across
models. For example, this good may be human capital as in Lucas (1988), the number
of varieties of products as in Romer (1990) or the quality of products as in Aghion and
Howitt (1992). Both sectors may use the same inputs according to technologies that may
be different.
The
This paper considers a two-sector model with overlapping generations, in which the in-
vestment goods are physical and human capital. The production of the final good and
the accumulation of human capital both uses quantities of physical and human capital as
inputs in proportions that are assumed to be exogenous as in Rebelo (1991). In standard
models, the returns to scale are constant in the output and the education sectors and the
growth rates of both sectors are equal. The level of the steady state income growth rate
depends on the fractions of physical and human capital allocated to both sectors and the
income shares of both investment goods.
More general specifications of the production functions for the final good and human
capital have been considered in the literature in order to study situations with increasing
returns. Mulligan and Sala-I-Martin (1993) study combinations of non-constant returns
to scale technologies in a neoclassical growth model that are consistent with balanced
growth in the long run. It is even possible to obtain increasing returns to scale in both
production functions if a non-reproducible factor such as land or raw labor (or a com-
bination of the two) is added to the output technology for example, and remain in a
competitive framework. This is the modeling approach we adopt in this paper and apply
it to a growth model with overlapping generations living for two periods. we consider
a general specification for production functions in the sense that we use Cobb-Douglas
functional forms in which the income shares of inputs may not sum up to one. Our ob-
jective is to identify the technological conditions yielding the highest income growth rate.
After determining the balanced growth condition along the stationary path, we compare
the growth rates yielded by the possible combinations of non-constant returns to scale
technologies that verify it, and give the conditions leading to the fastest steady state
growth rate.
The paper is organized as follows. Section 2 defines the model. Section 3 determines the
balanced growth condition. Section 4 studies the growth rates along the balanced growth
path. Section 5 presents an example derived from the Uzawa-Lucas model. Finally,
section 6 concludes.
1See Barro and Sala-I-Martin (1995) for a general treatment of two-sector growth models
1

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2The model
The model is an extension of the overlapping generations model of Allais (1947) and
Diamond (1965). The economy is closed and populated by overlapping generations, each
living for three periods. The generation in period t is populated by Nthouseholds and
the total population grows at a constant rate n. When young, the households work,
consume and invest a part of their income in both physical human capital which are
rented and used by the firms in the next period. When old, they consume the return of
their savings and die. In addition, each household owns a piece of land and a share of
the firms. They thus receive land rents and profits. As there is no land market, they
transmit their property rights over land to their children when they are old. As a result,
only the working generation owns land. Each household is owner of the firms and receives
interest on the capital rental. The firms buy inputs and produce the same single good
in perfectly competitive markets. The single final good produced in this economy can
either be consumed by the adult and the old generations or accumulated by the young
households as capital for future production. The only consumption good is measured in
units of final output.
2.1 Production technology
At each period the representative firm at the aggregate level produces a single good
under a technology with constant or non-constant (social) returns to scale. There are
three factors of production: physical capital, human capital and land. We assume that
the production function of the representative firm is given by
Yt= A(vKt)α(uHt)γLµ, 0 ? α,γ,µ,u,v ? 1(1)
where Ytis the output, Ktis physical capital, Htthe stock of human capital used by the
representative firm at time t, and A > 0 is a technological parameter. Physical capital is
assumed to be fully depreciated after one period. L is the land endowment of this economy,
which is assumed to be fixed over time and to enter the aggregate production function.2
Production uses a fraction v of physical capital and a fraction u of human capital. The
parameters α, γ and µ are the income shares or productivity elasticities of physical capital,
human capital and land respectively. Each of these parameters are assumed to be strictly
positive and strictly smaller than one. The problem of the firm is to maximize profits.
At time t, the firm inherits the stock of physical capital from the previous period and
has to decide upon the quantity of effective labor to hire. Therefore, an interior solution
corresponding to a maximum of profits exists if the production function is concave, i.e.,
if the returns to scale with respect to the reproducible factors are non increasing:
α + γ ? 1(2)
2The fixed factor could also be, for instance, raw labor or a combination of land and raw labor.
2

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2.2Human capital
Human capital is assumed to be a productive and a reproducible asset, whose accumu-
lation results from a technology that may be different from the one applying to physical
capital. Thus human capital may be an imperfect substitute for consumption. Like phys-
ical capital, it is assumed that human capital is homogenous. The production function
for the human capital accumulation is defined by
Ht+1= B[(1 − v)Kt]θ[(1 − u)Ht]η, 0 ? θ,η ? 1,(3)
where B > 0 is a technological parameter, θ and η are the elasticities of human capital
accumulation with respect to the past stocks of physical and human capital respectively.
The returns to scale of human capital accumulation are decreasing if θ+η < 1, constant if
θ+η = 1, and increasing if θ+η > 1. The stock of human capital at time t+1 is assumed
to depend on the inherited stocks of physical and human capital, which are imperfect
substitutes. If the fraction of the human capital input were higher than 1−u, then there
would be a positive externality in the sector of the human capital production. In other
words, social returns in the sector of human capital accumulation would be higher than
private returns. We assume that human capital fully depreciates after one period. If
θ < α, the sector of human capital accumulation is relatively intensive in human capital
while the output sector is relatively intensive in physical capital. This is the case in Uzawa
(1965) and Lucas (1988) where θ = 0 and α > 0. An interior solution for an optimal
choice of saving is obtained if the private marginal return to investment in physical capital
is decreasing, i.e.:
θ ? 1(4)
2.3 Preferences
The representative consumer maximizes a logarithmic utility function of the type
U(ct,dt+1) = lnct+ β lndt+1
(5)
subject to the following budget constraint,
ct+ st = wtuHt
Nt
+πtL
Nt
dt+1 = Rt+1st
Utility depends on consumption when young ctand on consumption when old dt+1. The
parameter β > 0 is the psychological discount factor. The adults supply inelastically one
unit of labor and earn wtuHt, where wtis the wage per unit of human capital and Ht
is the aggregate level of human capital. They also receive πtL/Ntas land rent. Their
income is allocated to consumption and saving, st, for future consumption. When old
agents spend all their saving and accrued interest on consumption. Rt+1is the interest
factor.
3