AAEC 3315
Agricultural Price Theory
CHAPTER 5
Theory of Production
The Case of One Variable Input
in the Short-Run
Objectives
To gain understanding of:
 Theory of Production

Production Curves
Total Physical Product
 Average Physical Product
 Marginal Physical Product




Law of Diminishing Physical Product
Stages of Production
Production Functions
Production Relationships

Definition: The technical relationship
between inputs & output indicating the
maximum amount of output that can be
produced using alternative amounts of
variable inputs in combination with one or
more fixed inputs under a given state of
technology.
Or, simply speaking, it is the technical
relationship between inputs & output
Product Curves
The Case of One Variable Input

X
TPP=Y
0.00
0.00
1.00
10.00
2.00
25.00
3.00
50.00
4.00
70.00
5.00
85.00
6.00
95.00
7.00
100.00
8.00
101.00
9.00
95.00
10.00
85.00
Total Physical Product (TPP) illustrates the technological or physical
relationship that exists between output
and one variable input, ceteris paribus

Starts increasing at an increasing
rate.

Continues to increase but at a
decreasing rate

Reaches the maximum, then
decreases

The functional form of a
production function is:
Y = f (X), where Y is the quantity
of output and X is the quantity of
input
Y
TPP
X
Product Curves
Y


The point where TPP
changes from
increasing at an
increasing rate to
increasing at a
decreasing rate is
called the Inflection
Points.
Points A, B, and C
Indicate total amount of
output produced at
each level of input use
Maximum Point
Y3
C
Y2
B
TPP
Inflection Point
Y1
A
X1
X2
X3
X
Product Curves (Cont.)

Average Physical
Product (APP) - shows
how much production, on
average, can be obtained
per unit of the variable input
with a fixed amount of other
inputs
X
TPP=Y
APP
0.00
0.00
1.00
10.00
10.00
2.00
25.00
12.50
3.00
50.00
16.67
4.00
70.00
17.50
5.00
85.00
17.00
6.00
95.00
15.83
7.00
100.00
14.29
8.00
101.00
12.63
9.00
95.00
10.56
10.00
85.00
8.50


Indicates average
productivity of the
inputs being used how productive is
each input level on
average
APP = Y / X
Drawing a line from
the origin which is
tangent to the TPP
curve gives APP max
Y
TPP
Y
X
APP
X
Product Curves (Cont.)

X
TPP=Y
MPP
0.00
0.00
1.00
10.00
10.00
2.00
25.00
15.00
3.00
50.00
25.00
4.00
70.00
20.00
5.00
85.00
15.00
6.00
95.00
10.00
7.00
100.00
5.00
8.00
101.00
1.00
9.00
95.00
-6.00
10.00
85.00
-10.00
Marginal Physical Product
(MPP) - represents the
amount of additional (i.e.,
marginal) output obtained
from using an additional unit
of variable input (X).

MPP = ΔY / ΔX = ∂Y/∂X
or the slope of the TPP curve.
Thus, MPP represents the
rate of change in output
resulting from adding one
more unit of input

Since MPP is the slope of
TPP, it reaches a maximum
at inflection point

It reaches zero at the
maximum point of TPP
Y
TPP
X
Y
APP
MPP
X
Product Curves
Y
X
TPP=Y
APP
MPP
0.00
0.00
1.00
10.00
10.00
10.00
2.00
25.00
12.50
15.00
3.00
50.00
16.67
25.00
4.00
70.00
17.50
20.00
5.00
85.00
17.00
15.00
6.00
95.00
15.83
10.00
7.00
100.00
14.29
5.00
8.00
101.00
12.63
1.00
9.00
95.00
10.56
-6.00
10.00
85.00
8.50
-10.00
TPP
X
Y
APP
MPP is negative
MPP
X
Relationships between
Product Curves








MPP reaches a maximum at inflection
point
MPP = 0 occurs when TPP is
maximum
MPP is negative beyond TPP max
Y
TPP
Drawing a line from the origin which is
tangent to the TPP curve gives APP
max
At point where APP is max, MPP
crosses APP (MPP=APP)
When MPP > APP, APP is increasing
When MPP = APP, APP is at a max
When MPP < APP, APP is decreasing
X
Y
The relationship between TPP, APP,
& MPP is very specific. If we have
COMPLETE information about one
curve, the other two curves can be
derived.
APP
MPP is negative
MPP
X
Law of Diminishing Marginal
Physical Product

Law of Diminishing Marginal Physical
Product: As additional units of one input
are combined with a fixed amount of other
inputs, a point is always reached where
the additional product received from the
last unit of added input (MPP) will decline

This occurs at the inflection point
Stages of Production:
Rational & Irrational



The stage I of the
production function is
between 0 and X1 units
of X.
In stage I:
 TPP is increasing
 APP is increasing
 MPP increases,
reaches a maximum
& decreases to APP
Stage I is an irrational
stage because APP is
still increasing
Y
I
TPP
X
Y
APP
0
X1
MPP
X
Stages of Production:
Rational & Irrational


The stage II of the production
function is between X1 and
X2 units of X.
In Stage II:
 TPP is increasing
 APP is decreasing
 MPP is decreasing and
less than APP, but still
positive

Y
I
TPP
II
X
Y
RATIONAL STAGE
BECAUSE TPP IS STILL
INCREASING
APP
0
X1
X2
MPP
X
Stages of Production:
Rational & Irrational


Stage III of the production
function is beyond X2 level X
In Stage III:
 TPP is decreasing
 APP is decreasing
 MPP is decreasing and
negative

Y
I
TPP
II
III
X
Y
IRRATIONAL STAGE
BECAUSE TPP IS
DECREASING
APP
0
X1
X2
MPP
X
A Hypothetical Production Function
Schedule
Total Physical Product Curve
Input
(X)
TPP
(Y)
APP
(Y/X)
0
0
0
1
10
10
10
2
25
12.5
15
3
50
16.67
25
4
70
17.5
20
5
85
17
15
6
95
15.83
10
7
100
14.29
5
8
101
12.63
1
9
95
10.55
-5
10
85
8.5
-10
MPP
(ΔTPP/ ΔX)
Output
Stage I
Stage II
Stage III
Input
APP/MPP
APP and MPP
0
1
2
3
4
5
Input
6
7
8
9
10
Effects of Technological Change



We know that the PF
gives the max amount
of output that can be
produced by a firm
using a given
technology.
The PF can shift over
time as a result of a
technological change
Technological change
refers to the introduction
of new technology that
increases output with
the same amount of
resources.
Y
TPP1
TPP
X
Elasticity of Production

The elasticity of production measures the degree of
responsiveness between output and input.
Q X
Q X
Q Q
MPP
Ep 






Q
X
X Q
X
X
APP

Q X

Using Calculus: Ep 
X
Q

Like any other elasticity, elasticity of production is
independent of units.

It measures the percentage change in production in
response to a percentage change in variable input.
A Hypothetical Production Function
A Mathematical Example



Consider a Production
Function
TP = X2 – 1/30X3,
where TP (Y) is quantity of
output and X is the quantity
of input.
AP = TP/X = X – (1/30)X2
Y
I
II
TPP
III
X
Y
MP = ∂TP/∂X
= 2X – (3/30)X2
= 2X – (1/10) X2
APP
0
MPP
X
A Hypothetical Production Function
A Mathematical Example
oGiven
Y
oTP = X2 – (1/30)X3,
oAP = TP/X = X – (1/30)X2
oMP = ∂TP/∂X = 2X – (1/10)X2
I
II
III
oAt what levels of X does the MP
reach its maximum?
oMP reaches its maximum
where ∂MP/∂X = 0
TPP
X
Y
oThat is, where
o
2 – (2/10)X = 0
oOr, 0.2 X = 2
oOr, X = 10
APP
0
10
MPP
X
A Hypothetical Production Function
A Mathematical Example
o Given
Y
oTP = X2 – (1/30)X3,
oAP = TP/X = X – (1/30)X2
oMP = ∂TP/∂X = 2X – (1/10)X2
I
II
TPP
III
o At what levels of X does the AP
reach its maximum?
o AP reaches its maximum
where ∂AP/∂X = 0
X
Y
o That is, where
o1 – (2/30)X = 0
oOr, (1/15) X = 1
oOr, X = 15
APP
0
10
15
MPP
X
A Hypothetical Production Function
A Mathematical Example
o Given,
Y
oTP = X2 – (1/30)X3,
oAP = TP/X = X – (1/30)X2
oMP = ∂TP/∂X = 2X – (1/10)X2
I
TPP
II
III
o At what levels of X does the TP
reach its maximum?
oTP reaches its maximum
where ∂TP/∂X = MP = 0
X
Y
oThat is, where
oMP = 2x – (1/10)X2 = 0
oUsing the quadratic
formula of
oX = 20
APP
0
10
15
20
MPP
X
A Hypothetical Production Function
A Mathematical Example
o Given
Y
oTP = X2 – (1/30)X3,
oAP = TP/X = X – (1/30)X2
oMP = ∂TP/∂X = 2X – (1/10)X2
I
TPP
II
III
o What is the range of X values for
Stage II?
o Stage II is the stage that
begins where AP is at its
maximum and ends where TP
is at its maximum.
X
Y
o Thus, the range of X values or
Stage II is 15 and 20.
APP
0
10
15
20
MPP
X
A Hypothetical Production Function
A Mathematical Example
o Given
Y
oTP = X2 – (1/30)X3,
oAP = TP/X = 2X – (1/30)X2
oMP = ∂TP/∂X = 2X – (1/10)X2
I
TPP
II
III
o At what level of X does the Law
of Diminishing Returns set in?
o It sets in where MP reaches
its maximum.
X
Y
o Thus at X = 10 the law of
Diminishing returns sets in.
APP
0
10
15
20
MPP
X
A Hypothetical Production Function
A Mathematical Example
o Given TP = X2 – (1/30)X3,
o At Y = 112.5 and X = 15, what is the elasticity of production?
o Applying
Ep 
Q X

X
Q
o Ep = (2X – (1/10) X2) * (X/Q)
o Ep = ((2*15) – (225/10)) * (15/112.5)
o Ep = (30 – 22.5) * (0.133)
o Ep = 7.5 * 0.133 = 0.997