A Golden Formula in Neoclassical‐Growth Models with Brownian‐Motion Shocks
| DOI | http://doi.org/10.1111/sjpe.12042 |
| Published date | 01 May 2014 |
| Date | 01 May 2014 |
| Author | Darong Dai |
A GOLDEN FORMULA IN NEOCLASSICAL-
GROWTH MODELS WITH BROWNIAN-
MOTION SHOCKS
Darong Dai*
ABSTRACT
In the paper, a Golden Formula, which does not depend on the specification of
production and preference functions, is established to reveal that time-average of
the growth rate of optimal capital accumulation will converge to a constant,
which is endogenously determined by relevant parameters, almost surely. The
Golden Formula naturally implies surprisingly interesting and also intrinsic eco-
nomic relations between some important macroeconomic variables; for example,
it serves as a direct bridge between the modified Golden Rule and the modified
Ramsey Rule. Furthermore, it indeed subsumes and hence substantially extends
the classical Golden Rule in deterministic theory.
II
NTRODUCTION
It is plausible to argue that national savings behavior plays a crucial role in
the process of economic growth. For example, Cole et al. (1992) insisted that
when savings behavior is investigated under different social norms or social
organizations, differences in growth rates across countries can be reasonably
explained. Moreover, some literatures (see, Anderson, 1990; Calder, 1990; and
among others) are interested in the high rates of household saving in Japan,
while the Chinese saving puzzle also has attracted extensive studies such as
Modigliani and Cao (2004), He and Cao (2007), and Wei and Zhang (2011).
Neoclassical theory of optimal capital accumulation and economic growth in
aggregate models has been well established (see, Phelps, 1961, 1965; Cass,
1965, 1966, 1972; Samuelson, 1965; Koopmans, 1967; Dai, 2012; and among
others) following the seminal papers of Ramsey (1928) and Solow (1956). The
current investigation follows Merton (1975), who has extended the classical
Ramsey and Solow models to stochastic cases, to establish a Golden Formula,
which constructs a direct bridge linking the modified Golden Rule and the
modified Ramsey Rule in neoclassical-growth models with Brownian-motion
shocks.
*Nanjing University
Scottish Journal of Political Economy, DOI: 10.1111/sjpe.12042, Vol. 61, No. 2, May 2014
©2014 Scottish Economic Society.
211
The major contribution of the paper is that we establish a Golden For-
mula, which says that time average of the growth rate of optimal capital
accumulation will converge to a constant, which is determined by relevant
model parameters, almost surely. The Golden Formula provides us with a
simple and exact characterization of the time-average and long-run behav-
iors of optimal capital accumulation. Moreover, it is robust in the sense
that it does not depend on the specification of production and preference
functions.
As is widely known, Golden Rule is the savings rule, which implies that
consumption level in the steady state of balanced path of capital accumulation
is maximized. However, the present study derives the Golden Formula in
Ramsey sense, and hence we call it Golden Formula rather than Golden Rule.
Indeed, Golden Rule is faced with the following severe challenges. Firstly,
steady states may not exist in many deterministic economies (see, Brock and
Mirman, 1972), while our analysis does not rely on the underlying existence;
secondly, some steady states are rather unstable and hence they cannot pass
the robust test (see, Brock and Mirman, 1972); thirdly, and also most impor-
tantly, the analysis based upon transitional dynamics should be paid more
attention because the explicit process of economic change or transition will
provide us with much more relevant information about the law of economic
development; finally, as is argued by Brock and Mirman (1972), the stable
steady state in deterministic models only provides us with a knife-edge case.
Thus, the Golden Formula overcomes all these shortcomings by sufficiently
incorporating stochastically dynamic economic processes into the neoclassical
analysis.
Corresponding to discrete-time models analyzed by the classic paper of
Brock and Mirman (1972) and Bayer and W€
alde (2011) have developed meth-
ods that allow to prove existence, uniqueness and stability of distributions
described by stochastic differential equations that are widely employed in con-
tinuous-time matching and savings models. Methodologically, Brock and Mir-
man (1972) use the classical stochastic stability theory of Markov chains
whereas Bayer and W€
alde (2011) expand their distributional analysis by using
the stability theory for Markov processes in continuous time. It is thus inter-
esting to compare the literature of Bayer and W€
alde (2011) with the current
study. First, similar to Bayer and W€
alde (2011), the current paper is also
motivated to understand the asymptotically dynamic properties of the sto-
chastic process governing the state variable capital stock per capita. Second,
the underlying Golden Formula studies long-run distribution of the growth
rate of optimal capital accumulation, which is hence different from that of
Bayer and W€
alde (2011) who directly focus on distributions of the state vari-
ables themselves. Thirdly, as you will see, the underlying Golden Formula
shows us a simple and also specific form of the long-run distribution, whereas
Bayer and W€
alde (2011) only demonstrate the existence, uniqueness and sta-
bility of the distributions. As a natural result, we can even analyze the inter-
nal relation between the modified Golden Rule and the modified Ramsey
Rule based upon the Golden Formula. Last but not least, it is easily seen that
212 DARONG DAI
Scottish Journal of Political Economy
©2014 Scottish Economic Society
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