Production
Production
an activity that creates value
Inputs for Production
raw materials, labor, land, capital, &
entrepreneurial or managerial talent.
Capital includes tools, machinery,
equipment, & physical facilities.
Production Function
Q = f(X1,X2,X3,X4,…,Xn)
where Q is the quantity of output that can
be produced with amounts of inputs,
X1,X2,X3,X4,…,Xn.
Short run
time period so short that the amounts of
some inputs can not be changed
For example, the quantity of plant & heavy
equipment can not be changed in a short
time period.
Long run
time period long enough for all inputs to be
changed
Fixed input
an input whose quantity can not be changed
in the short run
Variable input
an input whose quantity can be changed in a
short period of time
Examples: labor, raw materials
The scale of a firm’s operation is
determined by its fixed inputs.
We can look at the productivity of a variable
input given a fixed level of fixed input.
Marginal Product (MP)
of the variable input X
Discrete MP = ΔQ/ ΔX = ΔTP/ ΔX
Change in output resulting from a one-unit
change in the quantity of input
Continuous MP = dQ/dX = dTP/dX
Rate of change in total output as the usage
of the variable input increases by very
small amounts.
Graphical Interpretation of MP
Q = TP
The continuous MP is the slope of
the total product curve at a
particular point.
The discrete MP is the slope of the
line segment connecting 2 points on
the total product curve.
X
Example: Q = 21X + 9X2 – X3
X Q = TP
0
0
1
29
2
70
Example: Q = 21X + 9X2 – X3
Discrete MP
X Q = TP
ΔQ/ΔX
0
0
–
1
29
29
2
70
41
Example: Q = 21X + 9X2 – X3
Discrete MP
X Q = TP
ΔQ/ΔX
Continuous MP
dQ/dX =21+ 18 X – 3X2
0
0
–
–
1
29
29
36
2
70
41
45
Average Product (AP)
AP = Q / X = TP / X
Amount of product per unit of input
Can be calculated for variable or fixed inputs
Example: Q = 21X + 9X2 – X3
AP = Q / X = (21X + 9X2 – X3) / X
= 21 + 9X – X2
X = 1: AP = 29
X = 2: AP = 35
Graphical Interpretation of
AP = Q / X
The AP of a particular value
of X1 can be interpreted as
the slope of the line from the
origin to the corresponding
point on the curve.
Q = TP

Q1
0


X1
→
X
In this graph, we see that initially,
AP is increasing
Q = TP
0
X
and then decreasing
Q = TP
0
X
Principle of Diminishing Marginal Returns
As the amount of a variable input is
increased and combined with a specified
amount of fixed inputs, a point is
eventually reached where the resulting
increases in the quantity of output get
smaller & smaller.
In other words, as the amount of variable
input increases, eventually the MP of the
variable input falls.
Q = TP
MP
AP
Total Product,
Marginal Product, &
Average Product
Curves
incr marg
returns
X
MP
The TP curve gets
flatter as the slope of
TP falls.
AP
pt of dim
marg returns
Diminishing marginal
returns set in when
MP starts to fall (but
is still positive).
X
Q = TP
MP
AP
X
MP
AP
pt of dim
avg returns
X
Diminishing average
returns set in when
AP starts to fall.
(Remember that AP
is the slope of the
line from the origin to
the point on the TP
curve.)
Q = TP
MP
AP
dim total
returns
X
MP
AP
marginal returns
become negative
X
Diminishing total
returns set in when
the TP curve turns
downward and MP
becomes negative.
Isoquant
a curve showing all possible efficient
combinations of input that are capable of
producing a certain quantity of output
(Note: iso means same, so isoquant means
same quantity)
Isoquant for 100 units of output
100 units of output can be produced in
many different ways including
L1 units of labor & K1 units of capital,
L2 units of labor & K2 units of capital,
L3 units of labor & K3 units of capital, &
L4 units of labor & K4 units of capital.
Quantity of capital
used per unit of time
K1
K2
K3
100
K4
L1 L2
L3
L4
Quantity of labor
used per unit of time
Isoquants for different output levels
Quantity of capital
used per unit of time
As you move in a northeasterly
direction, the amount of output
produced increases, along with
the amount of inputs used.
125
100
50
Quantity of labor
used per unit of time
If you move out from the origin along a ray with
constant slope, the input combinations have a
constant capital-labor ratio.
Quantity of capital
used per unit of time
Each of the indicated points
uses one-third as much
capital as labor.
15
140
12
125
8
5
100
50
15
24
36
45
Quantity of labor used per unit of time
It is possible for an isoquant to have
positively sloped sections.
Quantity of capital
used per unit of time
In these sections, you’re
increasing the amounts of
both inputs, but output is
not increasing, because
the marginal product of
one the inputs is negative.
Quantity of labor used per unit of time
The lines connecting the points where the isoquants
begin to slope upward are called ridge lines.
Quantity of capital
used per unit of time
ridge lines
Quantity of labor used per unit of time
No profit-maximizing firm will operate at a point
outside the ridge lines, since it can produce the
same output with less of both outputs.
Quantity of
capital used per
unit of time
K2
B
A
K1
L1 L2
Notice, for example, that
since points A & B are on
the same isoquant, they
produce the same
amount of output.
However, point B is a
more expensive way to
produce since it uses
more capital & more
labor.
Quantity of labor used
per unit of time
Marginal rate of technical substitution
(MRTS)
The slope of the isoquant
The rate at which you can trade off inputs
and still produce the same amount of output.
For example, if you can decrease the
amount of capital by 1 unit while increasing
the amount of labor by 3 units, & still
produce the same amount of output, the
marginal rate of technical substitution is 1/3.
What is the MRTS or slope of the isoquant?
Quantity of capital
used per unit of
time
A
KA
KB
Consider 2 points A & B on the same isoquant.
Let’s divide the movement between A & B into 2 parts, from
A to C, & from C to B.
Moving from A to C, ΔQ = (ΔQ/ΔK) ΔK .
Moving from C to B, ΔQ = (ΔQ/ΔL) ΔL .
Moving from A to B, ΔQ = (ΔQ/ΔK) ΔK + (ΔQ/ΔL) ΔL
= MPK ΔK + MPL ΔL .
Since A & B are on the same isoquant, ΔQ = 0.
So, MPK ΔK + MPL ΔL = 0 .
B
MPK ΔK = - MPL ΔL .
C
ΔK/ΔL = - MPL/MPK
Q2
Q1
LA
LB
Quantity of labor
used per unit of time
slope of
isoquant
Marginal Rate of Technical Substitution
(MRTS)
or slope of an isoquant
ΔK/ΔL = - MPL/MPK
the negative of the ratio of the marginal
products of the inputs, with the input on the
horizontal axis in the numerator.
How does output respond to changes in
scale in the long run?
Three possibilities:
1. Constant returns to scale
2. Increasing returns to scale
3. Decreasing returns to scale
Constant returns to scale
Doubling inputs results in double the output.
Increasing returns to scale
Doubling inputs results in more than double
the output.
One reason this may occur is that a firm
may be able to use production techniques
that it could not use in a smaller operation.
Decreasing returns to scale
Doubling inputs results in less than double
the output.
One reason this may occur is the difficulty
in coordinating large organizations (more
paper work, red tape, etc.)
Graphs of Constant, Increasing, &
Decreasing Returns to Scale
Capital
Capital
Capital
150
150
100
50
Labor
Constant Returns to
Scale: isoquants for
output levels 50, 100,
150, etc. are evenly
spaced.
150
100
50
Labor
Increasing Returns to
Scale: isoquants for
output levels 50, 100,
150, etc. get closer &
closer together.
100
50
Labor
Decreasing Returns
to Scale: isoquants
for output levels 50,
100, 150, etc. become
more widely spaced.
Methods of estimating production functions
1. using statistical analysis of time series or
cross-sectional data.
2. based on experimentation or experience
with day-to-day operations.
A commonly used production function is the
Cobb-Douglas function
Q = AL1 K2 M3
where K is the quantity of capital, L is the quantity of
labor, & M is the quantity of raw materials. A, 1, 2, &
3 are parameters that depend on the specific case.
Also, 1, 2, & 3 are between 0 & 1.
If 1+ 2 + 3 = 1, we have constant returns to scale.
If 1+ 2 + 3 > 1, we have increasing returns to scale.
If 1+ 2 + 3 < 1, we have decreasing returns to scale.
Suppose Q = AL.5 K.2 M.5
Show that with 1+ 2 + 3 = .5 + .2 + .5 = 1.2 > 1, this production
function does have increasing returns to scale, by showing that
doubling inputs results in more than double the output.
Let Q’ be the output resulting from doubling the inputs.
Then Q’ = (A)(2L).5 (2K).2 (2M).5
= (A) (2.5 L.5) (2.2 K.2) (2.5 M.5)
= (A) (2.5) (2.2) (2.5)(L.5 K.2 M.5)
= (A) (2 .5 + .2 + .5)(L.5 K.2 M.5)
= 21.2 (A L.5 K.2 M.5)
> 2 (A L.5 K.2 M.5)  2Q
So doubling the inputs more than doubles the output.