09-09-2024
Production Theory
and Estimation
INTRODUCTION
• Production is the transformation of
resources into commodities over time.
• It is the process of transforming inputs
into output.
• Production is a flow concept- i.e. a rate of
output per unit of time. (As against stock
concept)
• The technological relationship between
inputs and output is referred to as
production function.
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Definitions:
1) A production function refers to the functional
relationship, between physical rates of inputs and
output of a firm, per unit of time under given
technology.
2) Production function refers to the functional
relationship between physical factor inputs and
output of a firm, per unit of time under given
technology.
3) “Production function is an equation, a table, or
graph showing the maximum output of a
commodity that a firm can produce per period of
time with each set of inputs.” Salvatore.
Characteristics of Pduction Function
• Flow Concept
• Physical Concept
• Technology assumed to be
constant (or is given)
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The Organization of Production
• Inputs
– Labor, Capital, Land
• Fixed Inputs
• Variable Inputs
• Short Run
– At least one input is fixed
• Long Run
– All inputs are variable
The Organization of Production
We may also divide inputs into the broad categories of
labor, materials and capital, each of which might include
more narrow subdivisions.
• Labor inputs include skilled workers (carpenters,
engineers) and unskilled workers (agricultural workers),
as well as the entrepreneurial efforts of the firm’s
managers.
• Materials include steel, plastics, electricity, water, and
any other goods that the firm buys and transforms into
final products.
• Capital includes land, buildings, machinery and other
equipment, as well as inventories. (P.192, Pindyck)
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Production Function
With Two Inputs
Q = f(L, K)
K Q
6 10 24 31 36 40 39
5 12 28 36 40 42 40
4 12 28 36 40 40 36
3 10 23 33 36 36 33
2 7 18 28 30 30 28
1 3 8 12 14 14 12
1 2 3 4 5 6 L
Production Function
With One Variable Input
1) Total Product: TP = Q = f(L)
TP
2) Average Product: APL =
L
TP
3) Marginal Product: MPL =
L
4) Production or MPL
Output Elasticity: EL = AP
L
Output ΔTP/TP ΔTP L ΔTP L ΔTP TP
---------- = ----- * ---- = ----- * --- = ----- ---
Elasticity ΔL/L TP L L TP L L
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Production Function
With One Variable Input
Total, Marginal, and Average Product of Labor, and Output Elasticity
L Q MPL APL EL
0 0 - - -
1 3 3 3 1
2 8 5 4 1.25
3 12 4 4 1
4 14 2 3.5 0.57
5 14 0 2.8 0
6 12 -2 2 -1
Production Function
With One Variable Input
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Production Function With One Variable Input
6.2 Production with One Variable
Input (Labor)
The Average Product of Labor Curve
In general, the average product of labor is given by
the slope of the line drawn from the origin to the
corresponding point on the total product curve.
The Marginal Product of Labor Curve
In general, the marginal product of labor at a point
is given by the slope of the total product at that
point.
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Law of Diminishing Returns
(Diminishing Marginal Product)
Holding all factors constant except one, the
law of diminishing returns says that:
• beyond some value of the variable input,
further increases in the variable input lead
to steadily decreasing marginal product of
that input
– e.g. trying to increase labor input
without also increasing capital will bring
diminishing returns
Law of Diminishing Marginal Returns- Defn 2
(Diminishing Marginal Product)P.198 Pindyck
The Law of Diminishing Marginal Returns
states that as the use of an input increases with
other inputs fixed, the resulting additions to
output will eventually decrease.
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Why marginal product curve rises and then fall? P.197, Pindyck
Consider a Television Assembly plant. Fewer than 10 workers might be
insufficient to operate the assembly line.
10-15 workers might be able to run the assembly line, but not very efficiently.
If adding a few more workers allowed the assembly line to operate much more
efficiently, then the marginal product of those workers would be very high.
This added efficiency might start diminishing once there are more than 20
workers.
The marginal product of the 22nd worker, for example, might still be very high
( and above average), but not as high as the marginal product of the 19th or
20th worker.
The Marginal product of the 25th worker might be lower still, perhaps equal
to the average product
With 30 workers, adding one more worker would yield more output, but not
very much more (MPL is positive but below average).
Once there are more than 40 workers, additional workers would
simply get in each others way and reduce actual output. So MPL
would be negative.
Three Stages of Production in Short Run
Stage I Stage III
APL, MPL
Stage II
APX
MPX X
Fixed input grossly Specialization and
underutilized; teamwork continue to Fixed input
specialization and result in greater capacity is
teamwork cause output when reached;
AP to increase additional X is used; additional X
when additional X fixed input being causes output to
is used properly utilized fall
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Optimal Use of the
Variable Input
Marginal Revenue
MRPL = (MPL)(MR)
Product of Labor
Marginal Resource TC
MRCL =
Cost of Labor L
Optimal Use of Labor: MRPL = MRCL
Optimal Use of the
Variable Input
Use of Labor is Optimal when: L = 3.50
L MPL MR = P MRPL MRCL
2.50 4 $10 $40 $20
3.00 3 10 30 20
3.50 2 10 20 20
4.00 1 10 10 20
4.50 0 10 0 20
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Optimal Use of the
Variable Input
Production With Two Variable
Inputs (p. 204 in Pindyck )
Isoquants show all possible combinations
of two inputs that can produce the same
level of output.
Firms will only use combinations of two
inputs that are in the economic region of
production, which is defined by the portion
of each isoquant that is negatively sloped.
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Production With Two
Variable Inputs - Deriving the Isoquants
Capital ( K )
Labour ( L )
Production With Two
Variable Inputs
Isoquants
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Production With Two
Variable Inputs
Economic
Region of
Production
Can you Identify the Ridge lines
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Production With Two
Variable Inputs
Marginal Rate of Technical Substitution
MRTS = -K /L = MPL / MPK
q q
dq = * dL * dK 0 See Pindyck
L K p. 207-208
q q
* dL () * dK
L K
dK MPL
q / L dK (-) ---- = -----
( ) =>
q / K dL dL MPK
Production With Two Variable Inputs
MRTS = |(-2.5/1)| = 2.5
Between points N & R,
MRTS = 2.5; between
R & S, MRTS = 1/2
and at R, it is 1, i.e
equal to slope of the
tangent.
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Production With Two Variable Inputs- Declining Marginal
Rate of Technical Substitution (MRTS) p. 207 Pindyck
Production With Two
Variable Inputs
Perfect Substitutes Perfect Complements
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Optimal Combination of Inputs
Isocost lines represent all combinations of
two inputs that a firm can purchase with
the same total cost.
C wL rK C Total Cost
w Wage Rate of Labor ( L )
r Cost of Capital ( K )
C w
K L => the equation to isocost line
r r
Optimal Combination of Inputs
Isocost Lines
AB C = $100, w = r = $10
A’B’ C = $140, w = r = $10
A’’B’’ C = $80, w = r = $10
AB* C = $100, w = $5, r = $10
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Optimal Combination of Inputs
MRTS = w/r
Optimal Combination of Inputs
As seen in the previous diagram, at the point of
tangency between Isoquant and Isocostline,
w
MRTS =
r
Also, MRTS = MPL
MPK
Thus, w MP L MP L MP K
OR
r MP K w r
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Optimal Combination of Inputs
Effect of a Change in Input Prices
Returns to Scale
Production Function Q = f(L, K)
Q = f(hL, hK)
If = h, then f has constant returns to scale.
If > h, then f has increasing returns to scale.
If < h, the f has decreasing returns to scale.
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Returns to Scale
Constant Increasing Decreasing
Returns to Returns to Returns to
Scale Scale Scale
Returns to Scale
Increasing Constant Decreasing
Returns to Returns to Returns to
Scale Scale Scale
As the scale of As the scale of
The forces of operation increases:
operation increases:
increasing
1) a greater 1) it becomes more
division of labour returns to scale and more difficult
and specialisation and decreasing to manage the firm
can take place. effectively ;
returns to scale
2) More productive 2) it becomes more
offset each and more difficult
and specialised
machinery can be other. to coordinate the
used. [The Ford various operations
Model-T's] and divisions of the
firm.[ L&T ]
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Empirical Production Functions
Cobb-Douglas Production Function
Q = AKaLb
Where, A is an initial condition, showing the amount of output when
K=1 &L=1
Estimated using Natural Logarithms
ln Q = ln A + a ln K + b ln L
Innovations in Production [for Global Competitiveness]
• Product Innovation Advantage US
• Process Innovation Favours Japan
• Product Cycle Model Getting shorter [News]
• Just-In-Time Production System Originated in Japan
• Competitive Benchmarking [1]
• Computer-Aided Design (CAD) US
• Computer-Aided Manufacturing (CAM)
• 3-D Printing
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Given:
Q = 98L- 3 L2
Price of the product = Rs.20
Wages = Rs. 400 per day.
How many units of labour to be used?
Answer: L = 13
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Mr. Kapoor, the owner and manager of the Fine duplicating
service located near a major university is contemplating keeping
her shop open after 4 P.M and until midnight. In order to do so, he
would have to hire additional workers. He estimates that the
additional workers would generate the following total output
(where each unit of output refers to 100 pages duplicated). If the
price of each unit of output is $10 and each worker hired must be
paid $40 per day, how many workers should Mr. Kapoor hire?
Workers hired 0 1 2 3 4 5 6
Total Product 0 12 22 30 36 40 42
Workers hired 0 1 2 3 4 5 6
Marginal product -
MRPL
Ans:5
1. Given the production function Q = 10√LK
a) Indicate whether this production function
exhibits constant, increasing or decreasing
returns to scale.
b) Does the production function exhibit
diminishing returns? If so, at when does the
law of diminishing return begin to operate?
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2. Given Q = 100 K0.5L0.5 ; C = $1000,
w= $30, r = $40. Determine the optimum
amount of K & L to be used. What is the
level of output?
3. Given the equation in Q2, find out the
optimum amount of K & L to be used to
produce 1,118 units of output.
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