A vintage model with endogenous growth and human capital

• Author: Jess Benhabib
• Published date: 01 March 2019
• Date: 01 March 2019
• DOI: http://doi.org/10.1111/ijet.12201

doi: 10.1111/ijet.12201

A vintage model with endogenous growth and human capital

Jess Benhabib∗

Wedevelop an endogenous growth model with v intages of physicalcapital where human capital

is endogenously produced. Under some simplifying assumptions, we can solve the model ana-

lytically and study its dynamics. Weprovide conditions under which a balanced growth path, in

the ratio of capital to human capital, exists and is stable, with the transition dynamics exhibiting

oscillatory echo effects associated with vintages.

Key wor ds vintage capital, endogenous growth

JEL classification 100, 130, 020

Accepted 21 August2018

1 Introduction

This paper builds on Benhabib and Rustichini (1994) to develop a vintage model by introducing

endogenous growth and human capital. As in Lucas(1988), human capital production is endogenous,

although we rule out external effects for simplicity. The production function for goods is given by

y=za1k1−ε

1+a2k1−ε

2+a3k1−ε

3+a4(H−h)1−ε1

1−ε,

where yis output and kiis the vintage of remaining capital produced iperiods ago. z>0 can be

an independent and identically distributed (i.i.d.) productivity shock or a constant. In this section

we abstract from time subscripts. For simplicity of exposition we assume capital vintages last three

periods, but the model trivially extends to vintages lasting nperiods and subject to one-hoss shay

depreciation after. Thus the constant elasticity of substitution (CES), production function is a vin-

tage aggregator and replaces the standard formulation of an aggregate capital stock as the sum of

depreciated past investments. The non-negative coefficients aimaysum to 1, as we will assume, but

are not necessarily restricted to be positive. Vintages are subject to depreciation, that is, wear and

tear depreciation by a factor μi, so for simplicity in the above μi=0 for i>4,but as noted, we

could have i>n. The depreciation factors μican also incorporate some aspects learning by doing

and need not be monotonic in i. Thus we have movedaway from exponential depreciation to a more

realistic and flexible depreciation scheme. Hrepresents the period endowment of human capital

with H−hallocated to the production of goods, and hallocated to the production of next period’s

human capital H=gh. The model can be generalized so that human capital is a non-rival input in

the production of Hor yin the sense of Romer (1990).

Instantaneous utility is assumed to be constant relative risk averse,

U(ct)=(1−ε)−1c(1−ε)−1,

∗Department of Economics, New YorkUniversity, New York, USA. Email: jb2@nyu.edu

International Journal of Economic Theory 15 (2019) 27–35 © IAET 27

International Journal of Economic Theory