2. Economic Growth and Convergence
1. The Production Function: Review
We begin our theoretical study of economic growth by considering how a country's technology
and factors of production–or factor inputs–determine its output of goods and services,
measured by real GDP. The relation of output to the technology and the quantities of factor
inputs is called a
production function
.
We will build a simplified model that has two factor inputs:
capital stock
,
K
, and labor,
L
. In this
model, capital takes a physical form, such as machines and buildings used by businesses. A more
complete model would include
human capital
, which embodies the effects of education and
training on workers' skills, and the effects of medical care, nutrition, and sanitation on workers'
health. In our simplified model, the amount of labor input,
L
, is the quantity of work-hours per
year for labor of a standard quality and effort. That is, we imagine that, at a point in time, each
worker has the same skill. For convenience, we often refer to
L
as the
labor force
or the number
of workers–these interpretations are satisfactory if we think of each laborer as working a fixed
number of hours per year.
We use the symbol
A
to represent the
technology level
. For given quantities of the factor inputs,
K
and
L
, an increase in
A
raises output. That is, a more technologically advanced economy has a
higher level of overall
productivity
. Higher productivity means that output is higher for given
quantities of the factor inputs.
Mathematically, we write the production function as
Key equation (production function):
(1)
One way to see how output,
Y
, responds to the variables in the production function–the
technology level,
A
, and the quantities of capital and labor,
K
and
L
–is to change one of the three
variables while holding the other two fixed. Looking at the equation, we see that
Y
is
proportional to
A
. Hence, if
A
doubles, while
K
and
L
do not change,
Y
doubles.
For a given technology level,
A
, the function
F(K, L)
determines how additional units of capital
and labor,
K
and
L
, affect output,
Y
. We assume that each factor is productive at the margin.
Hence, for given
A
and
L
, an increase in
K
–that is, a rise in
K
at the margin–raises
Y
. Similarly,
for given
A
and
K
, an increase in
L
raises
Y
.
The change in
Y
from a small increase in
K
is called the
marginal product of capital
, which we