Download - Production With Two Variable Inputs-f
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Production Function with two variable inputs
Rachita Gulati
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Production in the Long-Run 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 same level of output. Isoquant are also known as equal-product curves.
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Isoquants
0
20
40
60
80
100
120
140
160
0 10 20 30 40Labour
Ca
pit
al
An isoquant shows the
combinations of capital and labour that produce the
samesame level of output
240
358
Factor combination Labour Capital
A 10 100
B 20 60
C 30 50
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These Are Called IsoquantsThey slope down and to the right
Due to Marginal Rate of Technical Substitution
Higher isoquant represent higher output levelThey are convex to the origin They do not cross, although they are not necessarily
parallel
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MRTSLK would be the amount of capital that the firm would be willing to give up for an additional unit of labour.
MRTSLK = -K/L = MPL/MPK
MRTSKL would be the amount of labour that the firm would be willing to give up for an additional unit of capital.
MRTSKL = -L/K = MPK/MPL
Marginal Rate of Technical Substitution (MRTS)
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Law of Diminishing Marginal Rate of Technical Substitution:
for Isoquant Q = 52Combination K L
A 12 1B 8 2C 5 3D 3 4E 2 5
K L MRTSKL
-4 1 4 -3 1 3 -2 1 2 -1 1 1
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L = 1
K= -2
L = 1
K = -3
L= 1
K=- 4
A
B
C
DE
Downward sloping isoquant
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MRTS = K/L = - MPL/MPK
Units of KEmployed Output Quantity (Q)
18 37 60 83 96 107 117 127 12812 52 64 78 90 101 110 119 1208 37 52 64 73 82 90 97 1045 31 47 52 67 75 82 89 953 24 39 47 52 67 73 79 852 17 29 41 47 52 64 69 731 8 18 29 39 47 52 56 52
1 2 3 4 5 6 7 8Units of L Employed
52
52
52
5252
52
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Higher isoquant represents higher output
Units of KEmployed Output Quantity (Q)
8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 29 27 24 21 17
1 2 3 4 5 6 7 8Units of L Employed
Isoquant
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Economic region of production
Isoquants
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Economic Region of Production
Ridge Line
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Substituting inputsThere exists some degree of substitutability between inputs.
Different degrees of substitution:
Sugar
a) Perfect substitution b) Perfect complementarity
All other ingredients
Natural flavoring
Q
Q
Capital
Labor L1 L2 L3 L4
K1 K
2 K
3
K4
Cornsyrup
c) Imperfect substitution
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Isocost linesIsocost line is the locus of points of all the different
combinations of labour and capital that a firm can employ, given the total cost and prices.
Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.
If the price of labour is wage (w), the price of capital is interest (r), then total cost incurred by the firm is summarized as
C wL rK (M=Px. X+ Py. Y)
Isocost line
Budget line
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Isocost curvesIsocost 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
intercept slope
C wK L
r r
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The intercept of isocost line on the capital axis is the maximum amount of capital employed, when labour is not used in production process and is given by C/r.
The intercept on the labour axis is the maximum amount of labour used in the production process and is given by C/w.
Slope of the isocost lineSlope= Price of labour/Price of capital= w/r
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Changes in the Isocost Line
Changes in total costDecrease lead to a parallel,
inward shift in the isocost line.Increases lead to a parallel,
outward shift.
Changes in Price of labourA decreases in the price of
labour L rotates the isocost line counter-clockwise.
An increases rotates the budget line clockwise.
L
L
L
K
Rs. 500
Rs. 400
Rs. 300
10050
60 If price of labour falls from Rs. 4 to Rs. 3, isocost line shifts outwards.
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Producer’s Equilibrium: Optimal Combination of Inputs
MRTSLK = -K/L=w/r (slope of isoquant=slope of isocost line)
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Producer’s equilibriumProducer’s equilibrium shows lowest cost producing a given
level of output, where the isoquant corresponding to this output is tangent to the isocost line.
Thus, optimal combination of factor inputs depends on the relative prices of factor inputs and on the degree to which they can be substituted for one another.
This relationship can be stated as follows:MRTS=-K/L=MPL/MPK = PL/PK = w/r
(or MPL/PL= MPK/PK)
MPL/w= MPK/r
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Optimal level of inputsThe optimality conditions given in the previous slides ensure
that a firm will be producing in the least costly way, regardless of the level of output.
But how much output should the firm be producing?Answer to this depends on the demand for the product (like
in the one input case as well).
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Expansion pathThe locus of equilibrium points of isoquant with the
lowest possible isocost lineIt shows all the cost minimising input combinations for
various levels of output the firm could produce in the long run.
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Un
its
of
cap
ita
l (K
)
O Units of labor (L)
100
200
300
Expansion path
TC =Rs. 20 000
TC =Rs. 40 000
TC =Rs. 60 000
The long-run situation:both factors are variable
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Producer’s equilibrium vs. Consumer’s equilibrium
Producer’s equilibrium Producer’s theory Maximization of output Isoquant and isocost line Expansion path Equilibrium condition
MRTS= -K/L=w/r
Consumer’s equilibrium Consumer’s theory Maximization of satisfaction Indifference curve and
budget line Income consumption curve
or price consumption curve Equilibrium condition
MRS= -Y/X=Px/Py
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Returns to ScaleLet us now consider the effect of proportional increase in
all inputs on the level of output produced.To explain how much the output will increase, we will use
the concept of returns to scaleReturns to scale refers to the degree by which output changes
as a result of a given change in the quantity of all the factor inputs used in production.
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Returns to Scale
What happens to output when allall inputs are increased by a given percentage?
Three will be three situations:Increasing Returns
output increases by a largerlarger percentage Increase in factors by 10%, output increase by 20%
Constant Returnsoutput increases by the samesame percentage
Increase in factors by 10%, output increase by 10%Decreasing Returns
output increases by a smallersmaller percentage Increase in factors by 10%, output increase by 5%
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Returns to Scale
Units of LEmployed Output Quantity (Q)
8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17
1 2 3 4 5 6 7 8Units of K Employed
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In the previous table we are experiencing increasing returns to scale
Similarly, constant returns to scale and decreasing returns to scale are possible.
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Measurement of Returns to Scale
Coefficient of output elasticity:
EQ=
So if,EQ>1, increasing returns (proportionate increase in output is more
than proportionate increase in inputs)
EQ=1, constant returns (proportionate increase in output is in same
proportion to that of increase in inputs)
EQ<1, decreasing returns (proportionate increase in output is less
than proportionate increase in inputs)
percentage change in output
percentage change in all inputs
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Constant Returns to Scale
fig
Un
its o
f ca
pita
l (K
)
Units of labor (L)
100
200
300
400
500
a
b
cR
1K+1L (100)2K+2L (200)3K+3L (300)
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Increasing Returns to Scale (beyond point b)
fig
Un
its
of
cap
ita
l (K
)
Units of labor (L)
100
200
300
400
500
a
b
cR
600
1K+1L (100)1.7K+1.7L (200)2.2K+2.2L (300)
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Decreasing Returns to Scale (beyond point b)
fig
Un
its o
f ca
pita
l (K
)
Units of labor (L)
200
300
400
500
a
b
cR
1K+1L (100)2.5K+2.5L (200)4.5K+4.5L (300)
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Reasons for Increasing Returns to Scale:Division of labor (specialization)
Indivisibility of machinery or more sophisticated machinery justified
Economies of scale
Decreasing returns to scale can result from certain managerial inefficiencies:problems in communicationincreased bureaucracyIncreased use of fixed factor
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Production function as power function
Power function is the most frequently used type of production function in empirical work, even though it cannot exhibit two directions for marginal product on the same function.One reason for its popularity is that it can be readily transformed into a function with two or more independent variables:
mn
d3
c2
b1 VVVaVQ ...
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Production Function as Power FunctionProduction function with one variable input
Q = aLb
ifb > 1, Q increasing at increasing rate: MPL increasing
b = 1, Q increasing at constant rate: MPL constant
b < 1, Q increasing at decreasing rate: MPL decreasing
Major advantage of the power function is the fact that it can be transformed in a log-linear function
log Q = log a + b log L
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The Cobb-Douglas Production Function
A special case of power functions is Cobb-Douglas Production Function
Q = AKaLb
Estimated using natural logarithms
ln Q = ln A + a ln K + b ln L
Original version with constant returns to scale ( b + 1 - b = 1) introduced by Cobb in 1928.
Q = aLbK1-b,
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Reformulation by Cobb and Douglas:Q = aLbKc
b + c = 1, constant returns
b + c > 1, increasing returns
b + c < 1, decreasing returns
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Properties of the Cobb-Douglas function that have kept it so popular for 80 years
1. Both inputs have to be used simultaneously to get an output
2. Can exhibit different returns to scale
3. Allows to investigate MP for any factor while holding all others constant. So it is useful both in short-run and long-run analysis.
4. Elasticities are equal to the exponents b and c (constant in this formulation)
K
Qc
K
QMP
L
Qb
L
QMP
K
L
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Assignment problems1. If an estimated Cobb-Douglas production function is
Q = 10L0.8K0.6 (a) what are the output elasticities of capital and labour? If
the firm increases only the quantity of capital or only the labour used by 10%, by how much would output increase?
(c) What type of returns to scale does this production function indicate? If the firm increases at the same time both the quantity of capital and labour used by 10%, by how much would output increase?
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2. Suppose that the production function for a commodity is given by Q = 10L0.5K0.5 , where Q is the quantity of output, L is the labour units, and K is the capital units.
(a)Indicate whether this production function exhibits constant, increasing or decreasing returns to scale?
(b)Does the production function exhibits diminishing returns? If so, when does the law of diminishing returns begin to operate?
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3. The Royal furnishing manufacturers office furniture with the following production function:
Q = 20L0.1K0.9
the firm currently is producing with maximum efficiency and using 20 units of capital and 50 units of labour.
(a)What is the rate of output?
(b)What are the relative prices of capital and labour? What will be the actual price of labour and capital? Explain