Economic growth and economic development 81
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Introduction to Modern Economic Growth
dynamics by making the time unit as small as possible, i.e., by going to continuous time. While much of modern macroeconomics (outside of growth theory) uses
discrete time models, many growth models are formulated in continuous time. The
continuous time setup in general has a number of advantages, since some pathological results of discrete time disappear in continuous time (see Exercise 2.11).
Moreover, continuous time models have more flexibility in the analysis of dynamics
and allow explicit-form solutions in a wider set of circumstances. These considerations motivate a detailed study of both the discrete-time and the continuous-time
versions of the basic models.
Let us start with a simple difference equation
x (t + 1) − x (t) = g (x (t)) .
(2.29)
This equation states that between time t and t + 1, the absolute growth in x is given
by g (x (t)). Let us now consider the following approximation
x (t + ∆t) − x (t) ' ∆t · g (x (t)) ,
for any ∆t ∈ [0, 1]. When ∆t = 0, this equation is just an identity. When ∆t = 1,
it gives (2.29). In-between it is a linear approximation, which should not be too
bad if the distance between t and t + 1 is not very large, so that g (x) ' g (x (t))
for all x ∈ [x (t) , x (t + 1)] (however, you should also convince yourself that this
approximation could in fact be quite bad if you take a very nonlinear function g, for
which the behavior changes significantly between x (t) and x (t + 1)). Now divide
both sides of this equation by ∆t, and take limits to obtain
(2.30)
lim
∆t→0
x (t + ∆t) − x (t)
= x˙ (t) ' g (x (t)) ,
∆t
where throughout the book we use the “dot” notation
x˙ (t) ≡
∂x (t)
∂t
to denote time derivatives. Equation (2.30) is a differential equation representing the
same dynamics as the difference equation (2.29) for the case in which the distance
between t and t + 1 is “small”.
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