Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case



1 Introduction

In this chapter, we present the value function of a stochastic version of the Uzawa (1965)
and Lucas (1988) model of endogenous growth in discrete time. The externality of
human capital in goods production inherent to the model causes a difference between
the social planner’s solution and the market outcome. We pay attention to this fact
by treating both cases separately and by presenting both the social planner’s and the
representative agent’s value function. The chapter generalizes the results of Bethmann
(2002) where a deterministic version of this model was examined. Furthermore, we show
that the inefficiency of the decentralized economy can be overcome by introducing taxes
and subsidies on factor compensations.

The main feature of the Uzawa-Lucas model is the fact that the agents have to allo-
cate their human capital between two production sectors. On the one hand, there is a
goods sector where a single good usable for consumption and physical capital investment
is produced. This sector exhibits a production technology that uses human and physical
capital. On the other hand, there is a schooling sector where agents augment their stock
of human capital. Here, human capital is the only input factor. In short, agents have
to “learn or to do” (Chamley, 1993). In his seminal paper, Lucas (1988) argues that
the economy’s average level of human capital contributes to total factor productivity in
goods production. In a decentralized economy the individual’s accumulation of human
capital has no appreciable influence on this average level and agents are only compen-
sated for their respective factor supplies. This incentive structure leads to non-efficient
equilibria. Since agents are not able to coordinate their actions, their discounted utility
could be higher without making a single agent worse off. As a result, the solution for the
centralized economy deviates from that of the decentralized case.

The theoretical model considered here differs from that studied by Lucas (1988) in
two ways. First, there is our choice of the utility function. We assume logarithmic
preferences which imply that the constant intertemporal elasticity of substitution is equal
to one. This assumption reduces the number of parameters by one and simplifies the
calculations. Second, we assume discrete time where the two capital stocks depreciate
fully at the end of the period. This way the closed-form solution of the stochastic one
sector growth model with logarithmic preferences and full depreciation of physical capital
(cf. McCallum, 1989) is extended to the case with two capital goods.

The chapter is organized as follows. Section 2 introduces the model. Section 3
presents the value function in closed form as the solution to the social planner’s dynamic
optimization problem (DOP). In Section 4 we present the value function of the represen-
tative agent. Section 5 shows that the solutions are saddle path stable and determines
their time-series implications. Section 6 summarizes our results and concludes. Appen-
dix proves the uniqueness of the value functions found in the third and fourth section by
using an alternative method.

2 The model

We consider a closed economy populated by an infinite number of homogeneous, infinitely-
lived agents. The representative agent enters every period t with predetermined endow-
ments of human and physical capital, h
t and kt , respectively. Furthermore, there are
two sectors in the economy. Firms produce a single homogeneous good and a schooling
sector provides educational services.



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