1
Basic Notions of Production Functions
Lecture I
I.
Overview of the Production Function
A.
The production function is a technical relationship depicting the technical
transformation of inputs into outputs.
1.
The production function in and of itself is devoid of economic content.
2.
In the development of production functions, we are interested in
certain characteristics that make it possible to construct economic
models based on optimizing behavior.
B.
One way to write the production function is as a function map:
:
nm
f
RR
+
+
→
which states that the production function (
f
) is a function that maps
n
inputs
into
m
outputs.
By convention, we are only interested in positive input
bundles that yield positive output bundles.
C.
The first lecture will focus on the production function as a continuous function
as students have probably seen it in previous courses.
The next lecture will
develop the concept of the production function more rigorously.
II.
One Product, OneVariable Factor Relationships
A.
A commonly used form of the production function is the “closed form”
representation where the
total physical product
is depicted as a function of a
vector of inputs.
( )
yfx
=
where
y
is the scalar (single) output and
x
is a vector (multiple) inputs.
B.
Focusing for a moment on the single output case, we could simplify the above
representation to:
( )
12
yf
x
x
=
or we are interested in examining the relationship between
1
x
and
y
given
that all the other factors of production are held constant.
Using this
relationship, we want to identify three primary relationships:
1.
Total physical product
–which is the original production function.
2.
Average physical product
–defined as the average output per unit of
input.
Mathematically,
( )
f
x
y
APP
x
x
==
3.
Marginal physical product
–defined as the rate of change in
total
physical product
at a specific input level.
Mathematically,
( )
( )
()
dTPP
df x
dy
MPP
f
x
dx
dx
dx
′
=
=
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View Full DocumentAEB 6184 – Production Economics
Lecture I
Professor Charles B. Moss
Fall 2005
2
C.
Given these notions of a production function, we can introduce the classical
shape of the production function:
0
20
40
60
80
100
120
140
160
180
0
20
40
60
80
100
120
140
160
180
Nitrogen (lbs./acre)
Corn (bu./acre)
High Yield Function
Average Yield Function
Low Yield Function
This set of production functions are taken from Moss and Schmitz “Investing
in Precision Agriculture”.
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 Fall '09
 Staff
 Economics, Production Economics

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