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ECON 2450
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Lecture

Description

Neoclassical Growth Model - Solow
September 30, 2004 Environment
• Production Function : describes how, given the level of technology (E t, the economy’s pro-
ductive resources, capital ( tandlbr( L t are used to produce output,
Y = F (K ,E L ) (0.1)
t t t t
We assume that the production function has the following properties: (a) it exhibits constant
returns to scale (if you double inputs, you double output), (b) it is increasing in capital ( >
0), (c) exhibits diminishing returns to capital (as capital increases MPK falls).
• Savings/Investment : Consumers are assumed to consume a constant fraction, s< 1 ,ofeir
income (real GDP). Therefore total investment in this economy is,
It= sY t (0.2)
• Law of Motion for Capital :Acnft δ < 1 of the capital stock depreciates each
period. The total capital stock next period is then,
K t+1= K +tI −tδK t (0.3)
• Labor Force : The labor force is assumed to grow at a constant proportional rate per period.
Thus the labor force next period is,
Lt+1 =(1+ n)L t (0.4)
• Consumption: Whatever is not saved in this economy is consumed. Therefore investment
and consumption exhaust total output,
Ct+ I t Y t (0.5)
Equations (0.1) - (0.5) fully describe the environment of the neoclassical growth model.
The variables Y ,K ,I ,C are all aggregate or total variables. We are interested however in per
t t t t
worker variables because these re ﬂect the well being of the population. We use lower case letters
to denote the per worker variables: yt = Lt,k t= L , it= Lt, ct= Lt. Now we want to express
equations (0.1) - (0.5) in per worker terms. We simply divide both sides of these equations by .L
t
Then we have the following,
µ ¶
Y t K t
= F ,Et
L t Lt
It = s Yt
L t L t
L K K I
t+1 t+1 =(1 − δ) t+ t
L t+1 L t L t L t
1 Lt+1
=(1+ n)
Lt
C t It Y t
+ =
L t L t L t
Notice that in the law of motion I have multiplied and divided the term Kt+1 by Lt+1.he
Kt+1 Lt Lt+1
reason is that I want to create the variable Lt+1. But notice that I am also left with a term L t
which is equal to (1 + n). Substituting in this set of equations the lower case variables to which
they are equal, we have the following,
y = F (k ,E )
t t t
i = sy
t t
(1 + n)k =(1 − δ)k + i
t+1 t t
c + i = y
t t t
We can substitute t = sy t into the last two equation to be left with the following three
equations,
yt= F (k tE )t (0.6)
(1 + n)k =(1 − δ)k + sy (0.7)
t+1 t t
ct=(1 − s)y t (0.8)
Consider the transformed law of motion, equato in (0.7). W

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