Introduction to Production and Resource Use
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Transcript Introduction to Production and Resource Use
Technology
Beattie, Taylor, and Watts
Sections: 2.1-2.2, 5.1-5.2b
1
Agenda
The Production Function with One Input
Understand APP and MPP
Diminishing Marginal Returns and the
Stages of Production
The Production Function with Two Input
Isoquants
2
Agenda Cont.
Marginal Rate of Technical Substitution
Returns to Scale
Production Possibility Frontier
Marginal Rate of Product
Transformation
3
Production Function
A production function maps a set of
inputs into a set of outputs.
The production function tells you how to
achieve the highest level of outputs given a
certain set of inputs.
Inputs to the production function are also
called the factors of production.
The general production function can be
represented as y = f(x1, x2, …, xn).
4
Production Function Cont.
The general production function can be
represented as y = f(x1, x2, …, xn).
Where y is the output produced and is a
positive number.
Where xi is the quantity of input i for i = 1,
2, …, n and each is a positive number.
5
Production Function with One
Input
In many situations, we want to examine
what happens to output when we only
change one input.
This is equivalent to investigating the
general production function previously
given holding all but one of the variables
constant.
6
Production Function with One
Input Cont.
Mathematically we can represent the
production function with one input as
the following:
y = f(x) = f(x1) = f(x1|x2,x3,…,xn)
Suppose y = f(x1, x2, x3) = x1*x2*x3
Suppose that x2 = 3 and x3 = 4, which are
fixed inputs, then
y = f(x) = f(x1) = f(x1|3,4) = 12x1 =12x
7
Example of Production
Function
y = f(x) = -x3 + 60x2
Production Function
35000
30000
25000
20000
y
15000
10000
5000
0
0
10
20
30
40
50
60
8
APP and MPP
There are two major tools for
examining a production function:
Average Physical Product (APP)
Marginal Physical Product (MPP)
9
APP
The average physical product tells you
the average amount of output you are
getting for an input.
We define APP as output (y) divided by
input (x).
APP = y/x = f(x)/x
10
Example of Finding APP
Assume you have the following production
function:
y = f(x) = -x3 + 60x2
y x 60x
APP
x
x
APP x 2 60x
3
2
APP x(60 x)
11
Example of Finding Maximum
APP
To find the maximum APP, you take the
derivative of APP and solve for the x that
gives you zero.
From the previous example: APP = -x2 + 60x
dAPP d
2
max APP
x 60x
dx
dx
dAPP
2 x 60 0
dx
x 30
12
MPP
The marginal physical product tells you what
effect a change of the input will do to the
output.
In essence, it is the change in the output divided
by the change in the input.
MPP is defined as:
dy
MPP
f ' ( x)
dx
13
Example of Finding MPP
Assume you have the following production
function:
y = f(x) = -x3 + 60x2
dy d
MPP
x 3 60x 2
dx dx
2
MPP 3x 120x
MPP 3x( x 40)
14
Interpreting MPP
When MPP > 0, then the production
function is said to have positive returns
to the use of the input.
This occurs on the convex and the
beginning of the concave portion of the
production function.
In the previous example, this implies that
MPP > 0 when input is less than 40
(x<40).
15
Interpreting MPP Cont.
When MPP = 0, then we know that the
production function is at a maximum.
Setting MPP = 0 is just the first order
condition to find the maximum of the
production function.
In the example above, MPP = 0 when the
input was at 40.
16
Interpreting MPP Cont.
When MPP < 0, then the production
function is said to have decreasing
returns to the use of the input.
This occurs on the concave portion of the
production function.
In the previous example, this implies that
MPP < 0 when input is greater than 40
(x>40).
17
Example of APP and MPP
y = f(x) = -x3 + 60x2
APP and MPP
1500
1000
500
0
-500 0
10
20
30
40
50
60
APP
MPP
-1000
-1500
-2000
18
Law of Diminishing Marginal
Returns (LDMR)
The Law of Diminishing Marginal
Returns states that as you add
successive units of an input while
holding all other inputs constant, then
the marginal physical product must
eventually decrease.
This is equivalent to saying that the
derivative of MPP is negative.
19
Finding Where LDMR Exists
Suppose y = f(x) = -x3 + 60x2
To find where the LDMR exists is equivalent to
finding what input levels give a second order
condition that is negative.
dy
3x 2 120x
dx
dMPP d 2 y
2 6 x 120 6( x 20)
dx
dx
T o find where LDMR existsset theabove termnegative
dMPP
0 6( x 20) 0 ( x 20) 0
dx
x 20
MPP
20
Relationship of APP and MPP
When MPP > APP, then APP is rising
When MPP = APP, then APP is at a
maximum
When MPP < APP, then APP is declining
21
Relationship of APP and MPP
Cont.
f ( x)
x
dAPP d f ( x)
dx
dx x
dAPP xf ' ( x) f ( x)
dx
x2
APP
dAPP
xf ' ( x) f ( x)
0 when
0
dx
x2
dAPP
0 when xf ' ( x) f ( x) 0 * x 2
dx
dAPP
0 when xf ' ( x) f ( x)
dx
f ( x)
dAPP
0 when f ' ( x)
dx
x
dAPP
0 when MPP APP
dx
22
Stages of Production
Stage I of production is where the MPP
is above the APP, i.e., it starts where
the input level is 0 and goes all the way
up to the input level where MPP=APP.
To find the transition point from stage I to
Stage II you need to set the APP function
equal to the MPP function and solve for x.
23
Stages of Production Cont.
Stage II of production is where MPP is less
than APP but greater than zero, i.e., it starts
at the input level where MPP=APP and ends
at the input level where MPP=0.
To find the transition point from Stage II to Stage
III, you want to set MPP = 0 and solve for x.
Stage III is where the MPP<0, i.e., it starts at
the input level where MPP=0.
24
Graphical View of the
Production Stages
Y
TPP
Stage
I
Stage
II
Stage
III
MPP
APP
x
APP
MPP
x
25
Finding the Transition From
Stage I to Stage II
Suppose y = f(x) = -x3 + 60x2
dy
3 x 2 120x
dx
y 3 x 2 120x
APP
x 2 60x
x
x
T o find t he t ransit ion pointset MP P AP P and solve for x
APP MPP
MPP
x 2 60x 3 x 2 120x
2 x 2 60x 0
2 x( x 30) 0
x 0 or x 30
26
Finding the Transition From
Stage II to Stage III
Suppose y = f(x) = -x3 + 60x2
dy
3x 2 120x
dx
T o find the transition pointset MPP 0 and solvefor x
MPP 0
MPP
3x 2 120x 0
3x( x 40) 0
x 0 or x 40
27
Production Function with Two
Inputs
While one input production functions
provide much intuitive information
about production, there are times when
we want to examine what is the
relationship of output to two inputs.
This is equivalent to investigating the
general production function holding all but
two of the variables constant.
28
Production Function with Two
Inputs Cont.
Mathematically we can represent the
production function with one input as
the following:
y = f(x1,x2) = f(x1, x2|x3,…,xn)
29
Example of a Production Function with Two
Variables: y=f(x1,x2)=-x13+25x12-x23+25x22
5000.00
4500.00
4000.00
4500.00-5000.00
4000.00-4500.00
3500.00
3500.00-4000.00
3000.00
3000.00-3500.00
2500.00-3000.00
2500.00
2000.00-2500.00
2000.00
1500.00-2000.00
1500.00
1000.00-1500.00
500.00-1000.00
1000.00
24
0.00-500.00
18
500.00
12
0
24
22
20
18
16
14
12
10
6
8
4
6
2
0
0.00
30
Example 2 of a Production Function with Two
Variables: y=f(x1,x2)=8x11/4x23/4
80
70
60
70-80
50
60-70
50-60
40
40-50
30
30-40
20-30
20
10-20
10
6
4
5
4
3
2
7 6
2
1
0
10 9
8
8
0
10
0-10
0
31
Three Important Concepts for Examining
Production Function with Two Inputs
There are three important concepts to
understand with a production function
with two or more inputs.
Marginal Physical Product (MPP)
Isoquant
Marginal Rate Of Technical Substitution
(MRTS)
32
MPP for Two Input Production
Function
MPP for a production function with multiple
inputs can be viewed much like MPP for a
production function with one input.
The only difference is that the MPP for the
multiple input production function must be
calculated while holding all other inputs constant,
i.e., instead of taking the derivative of the
function, you take the partial derivative.
33
MPP for Two Input Production
Function Cont.
Hence, with two inputs, you need to calculate
the MPP for both inputs.
MPP for input xi is defined mathematically as
the following:
f ( x1 , x2 )
MPPxi
f xi
xi
f ( x1 , x2 )
MPPx1
f x1
x1
f ( x1 , x2 )
MPPx2
f x2
x2
34
Example of Calculating MPP
Suppose y = f(x1,x2) = -x13+25x12-x23+25x22
y f ( x1 , x2 ) x13 25x12 x23 25x22
f ( x1 , x2 )
MPPx1
3 x12 50x1
x1
f ( x1 , x2 )
MPPx2
3 x22 50x2
x2
35
Example 2 of Calculating MPP
Suppose y = f(x1,x2) = 8x11/4 x23/4
1
4
1
y f ( x1 , x2 ) 8 x x
3
4
2
3
3 3
4
f ( x1 , x2 ) 1 34 34
x
MPPx1
8 x1 x2 2 x1 4 x24 2 23
x1
4
x4
1
3
x2 4
2
x1
14
x1
6 1
x4
2
1
x1 4
6
x2
f ( x1 , x2 ) 3
MPPx2
8 x x
x2
4
1
4
1
1
4
2
1
4
1
6x x
1
4
2
36
Note on MPP for Multiple
Inputs
When the MPP for a particular input is
zero, you have found a relative extrema
point for the production function.
In general, the MPP w.r.t. input 1 does
not have to equal MPP w.r.t. input 2.
37
The Isoquant
An isoquant is the set of inputs that
give you the same level of output.
To find the isoquant, you need to set
the dependent variable y equal to some
number and examine all the
combinations of inputs that give you
that level of output.
An isoquant map shows you all the
isoquants for a given set of inputs.
38
Example of An Isoquant Map:
y = -x12+24x1-x22+26x2
24
21
18
15
12
9
6
24
20
16
12
8
4
0
3
300.00-350.00
250.00-300.00
200.00-250.00
150.00-200.00
100.00-150.00
50.00-100.00
0.00-50.00
-50.00-0.00
0
39
10
9
8
7
6
5
4
3
2
1
0
60-80
40-60
20-40
0-20
10
8
6
4
2
0
Example 2 of An Isoquant
Map: y = 8x11/4 x23/4
40
Finding the Set of Inputs for a General
Output Given y = -x12+24x1-x22+26x2
Suppose y = -x12+24x1-x22+26x2
We can solve the above equation for x2 in terms of
y and x1.
y x12 24x1 x22 26x2
x22 26x2 x12 24x1 y 0
If we define a 1, b 26, and c x12 24x1 y
b b 2 4ac
x2
2a
26 (26) 2 4(1)(x12 24x1 y )
x2
2(1)
26 676 4 x12 96x1 4 y
x2
2
x 2 13 169 x12 24x1 y
41
Question
From the previous example, does it make
economic sense to have both the positive and
negative sign in front of the radical?
No, only one makes economic sense; but which
one.
You should expect that you will have an inverse
relationship between x1 and x2.
This implies that for this particular function, you would
prefer to use the negative sign.
42
Finding the Set of Inputs for a
General Output Given y = 8x11/4 x23/4
Suppose y = 8x11/4 x23/4
We can solve the above equation for x2 in terms of
y and x1.
1
4
1
y f ( x1 , x2 ) 8 x x
3
4
2
x
3
4
2
y
1
4
1
8x
y4/3
x2
1/ 3
16x1
43
Marginal Rate of Technical
Substitution (MRTS)
The Marginal Rate of Technical
Substitution tells you the trade-off of
one input for another that will leave you
with the same level of output.
In essence, it is the slope of the isoquant.
44
Finding the MRTS
There are two methods you can find
MRTS.
The first method is to derive the isoquant
from the production function and then
calculate the slope of the isoquant.
The second method is to derive the MPP
for each input and then take the negative
of the ratio of these MPP.
45
Equivalency Between Slope of the
Isoquant and the Ratio of MPP’s
MRT S Slope of theisoquant
We know thatMP Px i
dx2
dx1
dy
dxi
dy
MP Px1 dx1
dy
MP Px 2
dx2
Since we are on t heisoquant the changein y for each MP Pis equal
MP Px1 dx2
MRTS
MP Px 2 dx1
46
Finding the MRTS Using the ratio of the MPP’s
Given y = -x12+24x1-x22+26x2
Suppose y = -x12+24x1-x22+26x2
y x12 24x1 x22 26x2
y
MPPx1
2 x1 24
x1
y
MPPx2
2 x2 26
x2
2 x1 24
MRTS
MPPx2
2 x2 26
MPPx1
47
Finding the MRTS Using the Slope of the
Isoquant Given y = -x12+24x1-x22+26x2
Suppose y = -x12+24x1-x22+26x2
From previouslywe found that theisoquant for theabovefunction was
26 676 4 x12 96x1 4 y
x2
2
x 2 13 169 x 24x1 y
2
1
1
2
x
2
13 169 x12 24x1 y
x1 x1
x
1
2 169 x12 24x1 y
x1
2
x 2
x1
x1 12
169 x
2
1
24x1 y
1
2
2 x 24
1
2
1
1
2
48
Finding the MRTS Using the Slope of the
Isoquant Given y = -x12+24x1-x22+26x2 Cont.
x 2
x1
x1 12
169 x
2
1
24x1 y
1
2
we know y x12 24x1 x22 26x2
x 2
x1
x 2
x1
x 2
x1
x1 12
169 x
24x1 x 24x1 x 26x2
169 x
24x1 x 24x1 x 26x2
2
1
2
1
169 x
26x2
x 2
x1 12
1
x1
2 2
x2 13
2
2
x1 12
x1 12
2
2
2
1
2
1
2
2
1
2
1
2
1
2
x 2 x1 12
x1
x2 13
49
Finding the MRTS Using the ratio of the MPP’s
Given y = Kx1x2
Suppose y = Kx1x2
y Kx1 x2
MPPx1
y
Kx1 1 x2
x1
MPPx2
y
Kx1 x2 1
x2
Kx1 1 x2
MRTS
MPPx
Kx1 x2 1
MPPx1
2
x1 1 x2 1
x11 x12
x
MRTS
2
x1
50
Finding the MRTS Using the Slope of the
Isoquant Given y = Kx1x2
Suppose y = Kx1x2
y Kx1 x2
x2
y
Kx1
y
x2
Kx1
1
1
y
x1
K
1
x 2
y
x1
x1 x1 K
1
x
y
2
x1
x1
K
1
1
x
y
2
x1
x1
K
51
Finding the MRTS Using the Slope of the
Isoquant Given y = Kx1x2 Cont.
But we know that y Kx1 x2
x
Kx1 x2
2
x1
K
x
Kx1 x2
2
x1
K
1
x
2
x1 x2 x1
x1
x1
1
x1
x
2
x2 x1
x1
x 2
x2
1
x2 x1
x1
x1
52
Returns to Scale
Returns to Scale examines what
happens to output when you change all
inputs by the same proportion, i.e.,
f(tx1,tx2).
There are three types of Returns to Scale:
Increasing
Constant
Decreasing
53
Increasing Returns to Scale
Increasing Returns to Scale are said to
exist when f(tx1,tx2)>tf(x1,x2) for t > 1.
This implies that as output is increasing,
the isoquants are getting closer together.
Suppose y = f(x1,x2) = x1x2
This implies that f(tx1,tx2) = tx1tx2 =t2x1x2
Comparing f(tx1,tx2) and tf(x1,x2) implies
f(tx1,tx2) = t2x1x2 >t f(x1,x2) = tx1x2,
because when t >1, t2 > t.
54
10
9
8
7
6
5
4
3
2
1
0
900-1000
800-900
700-800
600-700
500-600
400-500
300-400
200-300
100-200
0-100
10
8
6
4
2
0
Example Increasing Returns to
Scale: y = 10x1 x2
55
Constant Returns to Scale
Constant Returns to Scale are said to exist
when f(tx1,tx2)=tf(x1,x2) for t > 1.
This implies that as output is increasing, the
isoquants are the same distance apart.
Suppose y = f(x1,x2) = x1½ x2½
This implies that f(tx1,tx2) = (tx1)½ (tx2)½ =tx1x2
Comparing f(tx1,tx2) and tf(x1,x2) implies f(tx1,tx2)
= tx1x2 = t f(x1,x2) = tx1x2, because when t >1, t
= t.
56
10
9
8
7
6
5
4
3
2
1
0
10
8
6
4
2
0
Example Constant Returns to
Scale: y = 10x1½ x2½
95-100
90-95
85-90
80-85
75-80
70-75
65-70
60-65
55-60
50-55
45-50
40-45
35-40
30-35
57
Return to Scale Cont.
Decreasing Returns to Scale are said to exist
when f(tx1,tx2)<tf(x1,x2) for t > 1.
This implies that as output is increasing, the
isoquants are getting farther apart.
Suppose y = f(x1,x2) = x1 ¼ x2¼
This implies that f(tx1,tx2) = (tx1)¼ (tx2)¼
=t½x1¼x2¼
Comparing f(tx1,tx2) and tf(x1,x2) implies f(tx1,tx2)
= t½x1x2 < t f(x1,x2) = tx1¼x2¼, because when t
>1, t½ < t.
58
10
9
8
7
6
5
4
3
2
1
0
28-32
24-28
20-24
16-20
12-16
8-12
4-8
0-4
10
8
6
4
2
0
Example Decreasing Returns
to Scale: y = 10x1 ¼ x2¼
59
The Multiple Product Firm
Many producers have a tendency to
produce more than one product.
This allows them to minimize risk by
diversifying their production.
Personal choice.
The question arises: What type of
trade-off exists for enterprises that use
the same inputs?
60
Two Major Types of Multiple
Production
Multiple products coming from one
production function.
E.g., wool and lamb chops
Mathematically:
Y1, Y2, …, Yn = f(x1, x2, …, xn)
Where Yi is output of good i
Where xi is input i
61
Two Major Types of Multiple
Production Cont.
Multiple products coming from multiple
production functions where the
production functions are competing for
the same inputs.
E.g., corn and soybeans
62
Two Major Types of Multiple
Production Cont.
Mathematically:
Y1= f1(x11, x12, …, x1m)
Y2= f2(x21, x22, …, x2m)
Yn= fn(xn1, xn2, …, xnm)
Where Yi is output of good i
Where xij is input j allocated to output Yi
Where Xj x1j + x2j + … + xnj and is the
maximum amount of input j available.
63
Production Possibility Frontier
A production possibility frontier (PPF)
tells you the maximum amount of each
product that can be produced given a
fixed level of inputs.
The emphasis of the production possibility
function is on the fixed level of inputs.
These fixed inputs could be labor, capital, land,
etc.
64
PPF Cont.
All points along the edge of the production
possibility frontier are the most efficient use
of resources that can be achieved given its
resource constraints.
Anything inside the PPF is achievable but is
not fully utilizing all the resources, while
everything outside is not feasible.
65
Deriving the PPF
Mathematically
To derive the production possibility
frontier, you want to use the resource
constraint on the inputs as a way of
solving for one output as a function of
the other.
66
PPF Example
Suppose you produce two goods, corn
(Y1) and soybeans (Y2).
Also suppose your limiting factor is land
(X1) at 100 acres.
For corn you know that you have the
following production relationship:
Y1 = x1½
67
PPF Example Cont.
For corn you know that you have the
following production relationship:
Y2 = x2½
We know that 100 = x1 + x2.
68
Solving PPF Example
Mathematically
We know thefollowing:
1
2
1
Y1 x x1 Y12
1
2
2
Y2 x x2 Y22
x1 x2 100
Y12 Y22 100
Y22 100 Y12
1
2 2
1
Y2 100 Y
69
PPF Graphical Example
PPF
12
10
8
6
PPF
4
2
0
0
2
4
6
8
10
12
70
Marginal Rate of Product
Transformation (MRPT)
MRPT can be defined as the amount of
one product you must give up to get
another product.
This is equivalent to saying that the MRPT
is equal to the slope of the production
possibility frontier.
MRPT = dY2/dY1
Also known as Marginal Rate of Product
Substitution.
71
Find MRPT of the Following
PPF: Y2 = (100-Y12)½
Suppose Y2 = (100-Y12)½
1
2 2
1
Y2 100 Y
dY2 1
MRPT
100 Y12
dY1 2
dY2
MRPT
Y1 100 Y12
dY1
1
2
* 2Y1
1
2
0
72