Description
It also includes law of diminishing returns, properties of isoquants, marginal rate of technical substitution, isocost lines.
? Production theory forms the foundation for the
theory of supply
? Managerial decision making involves four types
of production decisions:
1. Whether to produce or to shut down
2. How much output to produce
3. What input combination to use
4. What type of technology to use
? Production involves transformation of inputs
such as capital, equipment, labor, and land into
output - goods and services
? Production theory can be divided into short
run theory or long run theory.
? The Long Run is distinguished from the short
run by being a period of time long enough for
all inputs, or factors of production, to be
variable as far as an individual firm is
concerned
? The Short Run, on the other hand, is a period
so brief that the amount of at least one input
is fixed
? The length of time necessary for all inputs to
be variable may differ according to the nature
of the industry and the structure of a firm
A production function is a table or a mathematical
equation showing the maximum amount of output
that can be produced from any specified set of inputs,
given the existing technology. The total product curve
for different technology is given below.
x
Q
Q = output
x = inputs
Q = f(X
1
, X
2
, …, X
k
)
where
Q = output
X
1
, …, X
k
= inputs
For our current analysis, let’s reduce the
inputs to two, capital (K) and labor (L):
Q = f(L, K)
? In the short run, capital is held constant.
? Average product is total product divided by
the number of units of the input
? Marginal product is the addition to total
product attributable to one unit of variable
input to the production process fixed input
remaining unchanged.
? MP = TP
N
– TP
N-1
labour Total product Average
product
Marginal
product
1 10 10 10
2 24 12 14
3 39 13 15
4 52 13 13
5 61 12.2 9
6 66 11 5
7 66 9.4 0
8 64 8 -2
? Marginal product at any point is the slope of
the total product curve
? Average product is the slope of the line
joining the point on the total product curve to
the origin.
? When Average product is maximum, the
slope of the line joining the point to the origin
is also tangent to it.
P: Maximum Average Product
Q & R : Same Average Product
? Both AP and MP first rise, reach a
maximum and then fall.
? MP = AP when AP is maximum.
? MP may be negative if Variable input is
used too intensively.
? Law of diminishing marginal productivity
states that in the short run if one input is
fixed, the marginal product of the variable
input eventually starts falling
Holding all factors constant except one, the law of
diminishing returns says that:
? As additional units of a variable input are
combined with a fixed input, at some point the
additional output (i.e., marginal product) starts
to diminish
?e.g. trying to increase labor input without also
increasing capital will bring diminishing returns
? Stage 1: Till average product becomes
maximum
? Stage 2: till MP =zero
? Stage 3: MP is negative
AP,MP
X
Stage I
Stage II
Stage III
AP
X
MP
X
? Section 2:
? All inputs are now considered to be variable
(both L and K in our case)
? How to determine the optimal combination of
inputs?
To illustrate this case we will use production isoquants.
An isoquant is a curve showing all possible combinations
of inputs physically capable of producing a given fixed
level of output.
fig
Units
of K
40
20
10
6
4
Units
of L
5
12
20
30
50
Point on
diagram
a
b
c
d
e
a
Units of labour (L)
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45 50
b
c
d
? Isoquant is a curve that shows the various
combinations of two inputs that will produce a
given level of output
? Slope of an isoquant indicates the rate at which
factors K and L can be substituted for each other
while a constant level of production is maintained.
? The slope is called Marginal Rate of Technical
Substitution (MRTS)
? Three general types of shapes that an
isoquant might have are:
• The isoquants are right angles, indicating
that inputs a and b must be used in fixed
proportions and therefore are not
substitutable
e.g Yeast and flour for a specific type of bread
b) Perfect Substitutes – in this case input a can be
substituted for input b at a fixed rate as indicated
by the straight line isoquants (which have a
constant slope and MRS)
Ie. Honey and brown sugar are often nearly perfect
substitutes, Natural gas and fuel oil are close
substitutes in energy production
? There is a different isoquant for every output rate the
firm could possibly produce with isoquants farther
from the origin indicating higher rates of output
? Along a given isoquant, the quantity of labor
employed is inversely related to the quantity of capital
employed ? isoquants have negative slopes
? Isoquants do not intersect. Since each isoquant refers
to a specific rate of output, an intersection would
indicate that the same combination of resources
could, with equal efficiency, produce two different
amounts of output
? Isoquants are usually convex to the origin ? any
isoquant gets flatter as we move down along the curve
c) Imperfect Substitutes – and the rate at which input
b can be given up in return for one more unit of
input a while maintaining the same level of output
(the MRS) diminished as the amount of input a
being used increases
Ie. In farming, harvestors and labour for harvesting
grain provide an example of a diminishing MRS,
and in general capital and labour are imperfect
substitutes.
? The absolute value of the slope of the isoquant is the
marginal rate of technical substitution, MRTS, between
two resources
? Thus, the MRTS is the rate at which labor substitutes for
capital without affecting output ? when much capital
and little labor are used, the marginal productivity of
labor is relatively great and the marginal productivity of
capital relatively small ? one unit of labor will substitute
for a relatively large amount of capital
Table 7.8 Input Combinations
for Isoquant Q = 52
Combination L K
A 6 2
B 4 3
C 3 4
D 2 6
E 2 8
AL AK MRTS
-2 1 2
-1 1 1
-1 2 1/2
0 2
? If labor and capital were perfect substitutes in
production, the rate at which labor substituted for
capital would remain fixed along the isoquant ? the
isoquant would be a downward sloping straight line
? Summary
? Isoquants farther from the origin represent higher rates
of output
? Isoquants slope downward
? Isoquants never intersect
? Isoquants are bowed toward the origin
? Anywhere along the isoquant, the marginal rate of
technical substitution of labor for capital equals the
marginal product of labor divided by the marginal
product of capital, which also equals the absolute
value of the slope of the isoquant
? MRTS = MP
L
/ MP
K
? Isocost lines show different combinations of inputs
which give the same cost
? At the point where the isocost line meets the vertical
axis, the quantity of capital that can be purchased
equals the total cost divided by the monthly cost of a
unit of capital ? TC / r
? Where the isocost line meets the horizontal axis, the
quantity of labor that can be purchased equals the
total cost divided by the monthly cost of a unit of labor
? TC / w
? The slope of the isocost line is given by
? Slope of isocost line = -(TC/r)/(TC/w) = -w/r
? The profit maximizing firm wants to produce its
chosen output at the minimum cost ? it tries to find
the isocost closest to the origin that still touches the
chosen isoquant.
Isocost Line - is a line that shows the various combinations
of two inputs that can be bought for a given dollar cost.
The equation for an isocost line is:
C =L. P
L
+K. P
K
Maximizing Output for a given cost
r
w
MP
MP
MRTS
K
L
LK
= =
Minimizing Cost subject to given Output
Q K L Q K L
490
15 99 470 14 100
500
15 100 500 15 100
Given the foll. data
If the price of labour is 5 per unit and price of capital is
10 per unit, is the combination of 15 capital and 100
labour the most efficient combination?
? If we imagine a set of isoquants representing each
possible rate of output, and given the relative cost of
resources, we can then draw isocost lines to determine
the optimal combination of resources for producing
each rate of output
? Expansion Path leads to Total Cost Curve
? An expansion path is a long-run concept
(because all inputs can change)
? Each point on the expansion path
represents a cost-minimizing
combination of inputs
? Given input prices, each point represents
a total cost of producing a given level of
output when the entrepreneur can
choose any input combination he or she
want
? If the relative prices of resources change, the least-
cost resource combination will also change ? the
firm’s expansion path will change
? For example, if the price of labor increases, capital
becomes relatively less expensive ? the efficient
production of any given rate of output will therefore
call for less labor and more capital
? Is large scale production more efficient than
small scale production for a certain market?
? Is a market better served by many
small firms or a few large ones?
? The returns to scale concept describes the
relationship between scale and efficiency.
? The returns to scale concept is an inherently
long run concept.
? Increasing returns to scale : a production
function for which any given proportional
change in all inputs leads to a more than
proportional change in output.
? Constant returns to scale : a production function for which a
proportional change in all inputs causes output to change by
the same proportion.
? Decreasing returns to scale : a production function for which
a proportional change in all inputs causes a less than
proportional change in output.
? Diminishing returns to scale is a short run
concept that refers to the case in which one
input varies while all others are held fixed.
? Decreasing returns to scale is a long run
concept that refers to the case in which all
inputs are varied by the same proportion.
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
600
a
b
c
R
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
600
a
b
c
R
700
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
a
b
c
R
? There are certain combinations of inputs that the
firm should not use in the long run no matter how
cheap they are (unless the firm is being paid to use
them)
? These input combinations are represented by the
portion of an isoquant curve that has a positive
slope
? A positive sloped isoquant means that merely to
maintain the same level of production, the firm
must use more of both inputs if it increases its use
of one of the inputs
? The marginal product of one input is negative, and
using more of that input would actually cause
output to fall unless more of the other input were
also employed.
? Ridge Lines – are lines connecting the points
where the marginal product of an input is equal to
zero in the isoquant map and forming the
boundary for the economic region of production
? Economic Region of Production – is the range in
an isoquant diagram where both inputs have a
positive marginal product. It lies inside the ridge
lines
? If both factors of production are increased by
proportion ?, and if new level of output Q*
can be expressed as a function of ? to any
power v, and the initial output ,
i.e. Q* = ?
V
Q
then the function is homogeneous and v is the
degree of homogeneity.
For example , check the function Q = 4L+3K
2
? If the production function is homogeneous ,
the expansion path is a straight line.
? Check the homogeneity of the following
functions:
? Q = 4L +3K
? Q= 4KL
? Q =4KL+K
? Three parameters: A, o, and |
? The Cobb-Douglas production function has
CRS if o+|=1
? The Cobb-Douglas production function has
increasing (decreasing) returns to scale if
o+|>(<)1
? If o=|=½, we have the square root
production function
q A L K
o |
= ·
? An isocline is a locus of points along which
MRTS is constant.
? An expansion path is also an isocline.
? An isocline is a straight line if the production
function is homogeneous
? Q =AK
?
L
?
If
?
+
? =
1 , we have CRS
> 1 , we have IRS
< 1, we have DRS
Check Q = L
2
K
2
doc_485172078.pptx
It also includes law of diminishing returns, properties of isoquants, marginal rate of technical substitution, isocost lines.
? Production theory forms the foundation for the
theory of supply
? Managerial decision making involves four types
of production decisions:
1. Whether to produce or to shut down
2. How much output to produce
3. What input combination to use
4. What type of technology to use
? Production involves transformation of inputs
such as capital, equipment, labor, and land into
output - goods and services
? Production theory can be divided into short
run theory or long run theory.
? The Long Run is distinguished from the short
run by being a period of time long enough for
all inputs, or factors of production, to be
variable as far as an individual firm is
concerned
? The Short Run, on the other hand, is a period
so brief that the amount of at least one input
is fixed
? The length of time necessary for all inputs to
be variable may differ according to the nature
of the industry and the structure of a firm
A production function is a table or a mathematical
equation showing the maximum amount of output
that can be produced from any specified set of inputs,
given the existing technology. The total product curve
for different technology is given below.
x
Q
Q = output
x = inputs
Q = f(X
1
, X
2
, …, X
k
)
where
Q = output
X
1
, …, X
k
= inputs
For our current analysis, let’s reduce the
inputs to two, capital (K) and labor (L):
Q = f(L, K)
? In the short run, capital is held constant.
? Average product is total product divided by
the number of units of the input
? Marginal product is the addition to total
product attributable to one unit of variable
input to the production process fixed input
remaining unchanged.
? MP = TP
N
– TP
N-1
labour Total product Average
product
Marginal
product
1 10 10 10
2 24 12 14
3 39 13 15
4 52 13 13
5 61 12.2 9
6 66 11 5
7 66 9.4 0
8 64 8 -2
? Marginal product at any point is the slope of
the total product curve
? Average product is the slope of the line
joining the point on the total product curve to
the origin.
? When Average product is maximum, the
slope of the line joining the point to the origin
is also tangent to it.
P: Maximum Average Product
Q & R : Same Average Product
? Both AP and MP first rise, reach a
maximum and then fall.
? MP = AP when AP is maximum.
? MP may be negative if Variable input is
used too intensively.
? Law of diminishing marginal productivity
states that in the short run if one input is
fixed, the marginal product of the variable
input eventually starts falling
Holding all factors constant except one, the law of
diminishing returns says that:
? As additional units of a variable input are
combined with a fixed input, at some point the
additional output (i.e., marginal product) starts
to diminish
?e.g. trying to increase labor input without also
increasing capital will bring diminishing returns
? Stage 1: Till average product becomes
maximum
? Stage 2: till MP =zero
? Stage 3: MP is negative
AP,MP
X
Stage I
Stage II
Stage III
AP
X
MP
X
? Section 2:
? All inputs are now considered to be variable
(both L and K in our case)
? How to determine the optimal combination of
inputs?
To illustrate this case we will use production isoquants.
An isoquant is a curve showing all possible combinations
of inputs physically capable of producing a given fixed
level of output.
fig
Units
of K
40
20
10
6
4
Units
of L
5
12
20
30
50
Point on
diagram
a
b
c
d
e
a
Units of labour (L)
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
0
5
10
15
20
25
30
35
40
45
0 5 10 15 20 25 30 35 40 45 50
b
c
d
? Isoquant is a curve that shows the various
combinations of two inputs that will produce a
given level of output
? Slope of an isoquant indicates the rate at which
factors K and L can be substituted for each other
while a constant level of production is maintained.
? The slope is called Marginal Rate of Technical
Substitution (MRTS)
? Three general types of shapes that an
isoquant might have are:
• The isoquants are right angles, indicating
that inputs a and b must be used in fixed
proportions and therefore are not
substitutable
e.g Yeast and flour for a specific type of bread
b) Perfect Substitutes – in this case input a can be
substituted for input b at a fixed rate as indicated
by the straight line isoquants (which have a
constant slope and MRS)
Ie. Honey and brown sugar are often nearly perfect
substitutes, Natural gas and fuel oil are close
substitutes in energy production
? There is a different isoquant for every output rate the
firm could possibly produce with isoquants farther
from the origin indicating higher rates of output
? Along a given isoquant, the quantity of labor
employed is inversely related to the quantity of capital
employed ? isoquants have negative slopes
? Isoquants do not intersect. Since each isoquant refers
to a specific rate of output, an intersection would
indicate that the same combination of resources
could, with equal efficiency, produce two different
amounts of output
? Isoquants are usually convex to the origin ? any
isoquant gets flatter as we move down along the curve
c) Imperfect Substitutes – and the rate at which input
b can be given up in return for one more unit of
input a while maintaining the same level of output
(the MRS) diminished as the amount of input a
being used increases
Ie. In farming, harvestors and labour for harvesting
grain provide an example of a diminishing MRS,
and in general capital and labour are imperfect
substitutes.
? The absolute value of the slope of the isoquant is the
marginal rate of technical substitution, MRTS, between
two resources
? Thus, the MRTS is the rate at which labor substitutes for
capital without affecting output ? when much capital
and little labor are used, the marginal productivity of
labor is relatively great and the marginal productivity of
capital relatively small ? one unit of labor will substitute
for a relatively large amount of capital
Table 7.8 Input Combinations
for Isoquant Q = 52
Combination L K
A 6 2
B 4 3
C 3 4
D 2 6
E 2 8
AL AK MRTS
-2 1 2
-1 1 1
-1 2 1/2
0 2
? If labor and capital were perfect substitutes in
production, the rate at which labor substituted for
capital would remain fixed along the isoquant ? the
isoquant would be a downward sloping straight line
? Summary
? Isoquants farther from the origin represent higher rates
of output
? Isoquants slope downward
? Isoquants never intersect
? Isoquants are bowed toward the origin
? Anywhere along the isoquant, the marginal rate of
technical substitution of labor for capital equals the
marginal product of labor divided by the marginal
product of capital, which also equals the absolute
value of the slope of the isoquant
? MRTS = MP
L
/ MP
K
? Isocost lines show different combinations of inputs
which give the same cost
? At the point where the isocost line meets the vertical
axis, the quantity of capital that can be purchased
equals the total cost divided by the monthly cost of a
unit of capital ? TC / r
? Where the isocost line meets the horizontal axis, the
quantity of labor that can be purchased equals the
total cost divided by the monthly cost of a unit of labor
? TC / w
? The slope of the isocost line is given by
? Slope of isocost line = -(TC/r)/(TC/w) = -w/r
? The profit maximizing firm wants to produce its
chosen output at the minimum cost ? it tries to find
the isocost closest to the origin that still touches the
chosen isoquant.
Isocost Line - is a line that shows the various combinations
of two inputs that can be bought for a given dollar cost.
The equation for an isocost line is:
C =L. P
L
+K. P
K
Maximizing Output for a given cost
r
w
MP
MP
MRTS
K
L
LK
= =
Minimizing Cost subject to given Output
Q K L Q K L
490
15 99 470 14 100
500
15 100 500 15 100
Given the foll. data
If the price of labour is 5 per unit and price of capital is
10 per unit, is the combination of 15 capital and 100
labour the most efficient combination?
? If we imagine a set of isoquants representing each
possible rate of output, and given the relative cost of
resources, we can then draw isocost lines to determine
the optimal combination of resources for producing
each rate of output
? Expansion Path leads to Total Cost Curve
? An expansion path is a long-run concept
(because all inputs can change)
? Each point on the expansion path
represents a cost-minimizing
combination of inputs
? Given input prices, each point represents
a total cost of producing a given level of
output when the entrepreneur can
choose any input combination he or she
want
? If the relative prices of resources change, the least-
cost resource combination will also change ? the
firm’s expansion path will change
? For example, if the price of labor increases, capital
becomes relatively less expensive ? the efficient
production of any given rate of output will therefore
call for less labor and more capital
? Is large scale production more efficient than
small scale production for a certain market?
? Is a market better served by many
small firms or a few large ones?
? The returns to scale concept describes the
relationship between scale and efficiency.
? The returns to scale concept is an inherently
long run concept.
? Increasing returns to scale : a production
function for which any given proportional
change in all inputs leads to a more than
proportional change in output.
? Constant returns to scale : a production function for which a
proportional change in all inputs causes output to change by
the same proportion.
? Decreasing returns to scale : a production function for which
a proportional change in all inputs causes a less than
proportional change in output.
? Diminishing returns to scale is a short run
concept that refers to the case in which one
input varies while all others are held fixed.
? Decreasing returns to scale is a long run
concept that refers to the case in which all
inputs are varied by the same proportion.
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
600
a
b
c
R
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
600
a
b
c
R
700
fig
0
1
2
3
4
0 1 2 3
U
n
i
t
s
o
f
c
a
p
i
t
a
l
(
K
)
Units of labour (L)
200
300
400
500
a
b
c
R
? There are certain combinations of inputs that the
firm should not use in the long run no matter how
cheap they are (unless the firm is being paid to use
them)
? These input combinations are represented by the
portion of an isoquant curve that has a positive
slope
? A positive sloped isoquant means that merely to
maintain the same level of production, the firm
must use more of both inputs if it increases its use
of one of the inputs
? The marginal product of one input is negative, and
using more of that input would actually cause
output to fall unless more of the other input were
also employed.
? Ridge Lines – are lines connecting the points
where the marginal product of an input is equal to
zero in the isoquant map and forming the
boundary for the economic region of production
? Economic Region of Production – is the range in
an isoquant diagram where both inputs have a
positive marginal product. It lies inside the ridge
lines
? If both factors of production are increased by
proportion ?, and if new level of output Q*
can be expressed as a function of ? to any
power v, and the initial output ,
i.e. Q* = ?
V
Q
then the function is homogeneous and v is the
degree of homogeneity.
For example , check the function Q = 4L+3K
2
? If the production function is homogeneous ,
the expansion path is a straight line.
? Check the homogeneity of the following
functions:
? Q = 4L +3K
? Q= 4KL
? Q =4KL+K
? Three parameters: A, o, and |
? The Cobb-Douglas production function has
CRS if o+|=1
? The Cobb-Douglas production function has
increasing (decreasing) returns to scale if
o+|>(<)1
? If o=|=½, we have the square root
production function
q A L K
o |
= ·
? An isocline is a locus of points along which
MRTS is constant.
? An expansion path is also an isocline.
? An isocline is a straight line if the production
function is homogeneous
? Q =AK
?
L
?
If
?
+
? =
1 , we have CRS
> 1 , we have IRS
< 1, we have DRS
Check Q = L
2
K
2
doc_485172078.pptx