Chapter 6
Production and Costs
The Case of One Variable Input
in the Short-Run
Steven Landsburg,
University of
Rochester
Copyright ©2008 by Thomson South-Western, a part of the Thomson Corporation. All rights
reserved.
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
Factors of Production

Inputs
 Fixed Inputs - factors must be maintained (i.e. paid
for, kept up, etc.) whether production occurs or not
(ex. - land, buildings, heavy machinery, etc.)


Variable inputs - factors that vary as the output
level changes (ex. - labor, fertilizer, seed, etc.)
Assumption
 For this section, assume that only one variable
input (e.g., labor) is required to produce a good.
Production Relationship
The Case of One Variable Input

Total Product (TP) - illustrates
the technological or physical
relationship that exists between
output and one variable input,
ceteris paribus



Y
Starts increasing at an
increasing rate.
Continues to increase but at a
decreasing rate
TP
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
Reaches the maximum, then
decreases
The functional form of a production function is given by:
Y = f (X), where Y is the quantity of output and X is the quantity of input
X
Product Curves

The point where TP
changes from increasing at
an increasing rate to
increasing at a decreasing
rate is called the Inflection
Point.
Y
Maximum Point
Y3
C
Y2
B
TP
Inflection Point

Points A, B, and C Indicate
total amount of output
produced at each level of
input use
Y1
A
X1
X2
X3
X
Product Curves (Cont.)

Average Product (AP) - shows how much production, on average, can
be obtained per unit of the variable input with fixed amounts of other
inputs

Indicates average productivity of the inputs being used how productive is each input level on average
AP = Y / X


Drawing a line from the origin which is tangent to the TP
curve gives maximum AP
Marginal Product (MP) - represents the amount of additional (i.e.,
marginal) output obtained from using an additional unit of the variable
input (X).
 MP = ΔY / ΔX = ∂Y/∂X, or the slope of the TP curve. Thus, MP
represents the rate of change in output resulting from adding one
more unit of input
 Since MP is the slope of TP, it reaches a maximum at inflection point
 It reaches zero at the maximum point of TP
Product Curves (Cont.)
X
Y = TP
AP=Y/X
0
0
0
1
10
10.0
2
25
12.5
3
50
16.7
4
70
17.5
5
85
17.0
6
95
15.8
7
100
14.3
8
101
12.6
9
95
10.6
10
85
8.5
Y
TP
Y
X
AP
X
Product Curves (Cont.)
MP=∂Y/∂X
X
Y = TP
AP=Y/X
0
0
0
1
10
10.0
10
2
25
12.5
15
3
50
16.7
25
4
70
17.5
20
5
85
17.0
15
6
95
15.8
10
7
100
14.3
5
8
101
12.6
1
9
95
10.6
-6
10
85
8.5
-10
Y
TP
X
Y
AP
MPP
X
Product Curves
MP=∂Y/∂X
X
Y = TP
AP=Y/X
0
0
0
1
10
10.0
10
2
25
12.5
15
3
50
16.7
25
4
70
17.5
20
5
85
17.0
15
6
95
15.8
10
7
100
14.3
5
8
101
12.6
1
9
95
10.6
-6
10
85
8.5
-10
Y
TP
X
Y
AP
MPP is negative
MP
X
Relationships between
Product Curves








MP reaches a maximum at inflection
point
MP = 0 occurs when TP is maximum
MP is negative beyond TP max
Y
TP
Drawing a line from the origin which is
tangent to the TP curve gives AP max
At point where AP is max, MP crosses AP
(MP=AP)
Y
X
When MP > AP, AP is increasing
When MP = AP, AP is at a max
When MP < AP, AP is decreasing
The relationship between TP, AP, & MP is
very specific. If we have COMPLETE
information about one curve, the other two
curves can be derived.
AP
MPP is negative
MP
X
Law of Diminishing Marginal
Physical Product

Law of Diminishing Marginal 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 (MP) 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:
 TP is increasing
 AP is increasing
 MP increases, reaches a
maximum & decreases to
AP
Y
I
TP
X
Y
AP

Stage I is an irrational stage
because AP is still increasing
0
X1
MP
X
Stages of Production:
Rational & Irrational


The stage II of the production
function is between X1 and
X2 units of X.
In Stage II:
 TP is increasing
 AP is decreasing
 MP is decreasing and less
than AP, but still positive
 Rational stage, because
TP is still increasing
Y
I
TP
II
X
Y
AP
0
X1
X2
MP
X
Stages of Production:
Rational & Irrational

Stage III of the production
function is beyond X2 level
Y
I
TP
II

In Stage III:
 TP is decreasing
 AP is decreasing
 MP is decreasing and
negative

Stage III is a irrational
stage because TP is
declining
III
X
Y
AP
0
X1
X2
MP
X
A Hypothetical Production Function
Schedule
Total Physical Product Curve
Input
(X)
TP
(Y)
AP
(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
MP
(ΔY/ ΔX)
Output
Stage I
Stage II
Stage III
Input
APP/MPP
APP and MPP
0
1
2
3
4
5
Input
10
85
8.5
-10
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.
Y
TP’
TP
The PF can shift over time as
a result of a technological
change
X

Technological change refers
to the introduction of new
technology that increases
output with the same amount
of resources.