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8/10/2019 ME Production & Cost Theory
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Production Theory and Analysis
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Production
Productionrefers to the transformation of inputs or
resources into outputs of goods and services
Creation of utility
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Characteristics of goods & services to be
classified as Production are:
Created by human labor and capital
Satisfy human wants directly or indirectly Are comparatively scarce and have economic
value
Have a definite monetary price/cost
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EnterpriseCapitalLaborLand
Immobile
Passive
Heterogeneous
Active
Mobile
Variableproductivity
Structures
Equipment
Cap.goodsMoney
Innovative function
Risk
Decision making
Factors of Production
INPUTS
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INPUTS
CAPITAL
EntrepreneurWorkersLand &
Structures
LABOR
Machinery
plant &
equipment
Natural
Resources
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Factors of Production
Inputs
Fixed Inputs
Variable Inputs
Short Run- At least one input is fixed
Long Run- All inputs are variable
The length of long run depends on industry.
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Level & Scale of Production
Level of productioncan be altered changing the
proportion of variable inputs
Output = Fixed inputs + Variable inputs
Scale of productioncan be altered by changing the
supply of all the inputs only in the long run
Output = Total inputs(variable inputs)
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Concept of Product
Total Product- total volume of goods producedduring a specific period of time
Average Product- the per unit product of avariable factor
Marginal Product- the rate at which totalproduct increases / addition to total productresulting from a unit increase in the quantity of thevariable factor
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Production Function With Two Inputs
Input & output are measured in physical units
Assumption- Technology is constant the during analysis period
- All units of L & K are homogenous
K Q6 10 24 31 36 40 395 12 28 36 40 42 40
4 12 28 36 40 40 363 10 23 33 36 36 332 7 18 28 30 30 281 3 8 12 14 14 12
1 2 3 4 5 6 L
Q = f(L, K)
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Production Function with Two Inputs
Discrete Production Surface
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Production Function with Two Inputs
Continuous Production Surface
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Production Function with
One Variable Input
Total Product TP = Q = f(L)
Marginal Product MPL=
TPL
Average Product APL=TP
LProduction orOutput Elasticity
Q/QL/L
Q/ LQ/L
== =ELMPL
APL
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Production Function with
One Variable Input
L Q
0 0
1 3
2 8
3 12
4 14
5 14
6 12
Total, Marginal, and Average Product of Labor, and Output Elasticity
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Production Function with
One Variable Input
L Q MPL APL EL
0 0 - - -1 3 3 3 1
2 8 5 4 1.25
3 12 4 4 14 14 2 3.5 0.57
5 14 0 2.8 0
6 12 -2 2 -1
Total, Marginal, and Average Product of Labor, and Output Elasticity
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Production Function
with One Variable Input
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-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
6
A
B
C
D E
F
A
B CD
EF
TotalProduct
Marginal
& Average
Product
Labor
Labor
Production Function with One Variable Input
TP
MP
AP
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-2
-1
0
1
2
3
4
5
0 1 2 3 4 5 6 7
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7
6
A
B
C
D E
F
B C
AD
EF
I
TotalProduct
Marginal
& Average
Product
Labor
Labor
The Law of Diminishing Returns &
Stages of Production
Stage I of Labor Stage II of Labor Stage III of Labor
TP
MP
AP
G
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Optimal Use of the Variable Input
Marginal Revenue
Product of Labor
MRPL= (MPL)(MR)
Marginal ResourceCost of Labor
MRCL=TCL
Optimal Use of Labor MRPL= MRCL
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L MPL MR = P
2.50 4 $10
3.00 3 10
3.50 2 10
4.00 1 10
4.50 0 10
Optimal Use of the Variable Input
Assumption : Firm hires additional units of labor at constantwage rate = $20
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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
Use of Labor is Optimal When L = 3.50
Optimal Use of the Variable Input
Assumption : Firm hires additional units of labor at constantwage rate
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Optimal Use of the Variable Input
2.5 3.0 3.5 4.0 4.5
40
30
20
10
0
MRCL= w = $20
dL= MRPL
Units of Labor Used
$
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Exercise
The marginal product of labor equation for a firm
is given by: MPL= 10(K/L)0.5
Currently the firm is using 49 units of capital and
100 units of labor. Capital usage is fixed, but labor
can be varied. If the price of labor is $20 per unit
and the firms' output sells for $4, is the firm
producing efficiently in the short run? If not,explain and determine the optimal rate of labor
input.
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MRPL = MRCL = w
28 20 Not efficient
L = 196
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Production With Two Variable Inputs
Isoquants show combinations of two inputsthat can produce the same level of output.
K
Q
6 10 24 31 36 40 395 12 28 36 40 42 404 12 28 36 40 40 36
3 10 23 33 36 36 332 7 18 28 30 30 281 3 8 12 14 14 12
1 2 3 4 5 6 L
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Isoquants
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Economic Region of ProductionFirms will only use combinations of two inputs that are in
the economic region of production.
W
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MRTS = -(-2.5/1) = 2.5
Marginal Rate of Technical Substitution
Absolute value of the slope of isoquant is called the MRTS
K
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Production With Two Variable Inputs
Perfect Substitutes Perfect Complements
6
4
2
2 4 6 8 10 12
6
4
2
2 4 6
CapitalCapital
Labor Labor0 0
-1K
2L
2K
1L
C
A
B
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10
8
6
4
2
2 4 6 8 10
Capital
Labor
1K
1L
AB C = $100, w = r = $10A
B
Isocost Lines
slope = -w/r = -1
vertical intercept = 10
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4 8 10 12 14 16 20
14
108
4
A
A
A
BB B B*
0 Labor
Capital
Isocost Lines
Isocost Lines
AB C = $100, w = r = $10
AB C = $140, w = r = $10
AB C = $80, w = r = $10
AB* C = $100, w = $5, r = $10
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Optimal Combination of Inputs for
Minimizing costs or Maximizing output
MRTS = w/r
Isocost Lines
AB C = $100, w = r = $10
AB C = $140, w = r = $10
AB C = $80, w = r = $10
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Product lines
A product line shows the movementfrom one isoquant to another as we
change both factors or a single factor.
The product line describes thetechnically possible alternative paths of
expanding output and what path
actually be chosen by the firm willdepend on the prices of factors.
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The product curve passes through the origin if
all factors are variable. If only one factor isvariable (the other being kept constant) the
product line is a straight line parallel to the
axis of variable factor. The K/L ratio
diminishes along the product line.
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Optimal input combinations:
Slope of isoquant = Slope of isocost line
(absolute)Slope of isoquant = (absolute) Slope of isocost line
MRTS = w
r
Since MRTS = MPL/ MPK
MPL = w
MPK r
MPL = MPKw r
If MPL = 5, MPK =4, and w = r
MPL > MPK
w r
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Profit Maximization
MRP(input) = MRC(input)
with constant input prices
MRP(input) = input price
To maximize Profits:
MRPL= w = (MPL)(MR)
MRPK= r = (MPK)(MR)
MPL = MPKw r
MPL = w
MPK r
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Optimal Combination of Inputs
Effect of a Change in Input Prices
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Returns to Scale
Production Function Q = f(L, K)
Q = f(hL, hK)
If = h, constant returns to scale.
If > h, increasing returns to scale.
If < h, decreasing returns to scale.
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Returns to Scale
ConstantReturns to
Scale
IncreasingReturns to
Scale
DecreasingReturns to
Scale
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Empirical Production Functions
Cobb-Douglas Production Function
Q = AKaLb
If a + b = 1, constant returns to scale.
If a + b > 1, increasing returns to scale.If a + b
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Q=ALa Kb
Where Q= Manufacturing output
L= quantity of labour employed
K= quantity of capital employed
A, a, b are constant or parameter of funcions.
1. the sum of exponents of factors i.e. a+b measures returns toscale. If
If a + b = 1, constant returns to scale.
If a + b > 1, increasing returns to scale. If a + b
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2. Average and marginal product of factors
depend upon the ratios in which factors are
combined for the production of a
commodity.
APL=Q/L=ALaKb/L=A(K/L)b
Thus ,average product of labour depends on
the ratios of the factor(K/L) and does not
depend upon the absolute quantities of thefactor used.
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CobbDouglas Production Function
and Output Elasticities of Factors
The exponents of labour and capital in
cobb-douglas production function measure
output elasticities of labour and capital.
Output elasticity of a factor refers to the
relative or percentage change in output
caused by a given percentage change in a
variable factor, other factors and inputs
remaining constant.
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Elasticity of Technical Substitution (between
factors) =
Proportionate change in factor proportions
K/L
Proportionate change in MRTS between
labour and capital
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Exercise
Do the following production functions have constant,
increasing or decreasing returns of scale? ( K, L, M
are inputs)
a. Q = 0.5X + 2Y + 40Z
b. Q = 3L + 10K + 500
c. Q = K0.3L0.5
d. Q = 4A2+ 6B2+ 8AB
e. Q = 10L 0.5K 0.6
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A. constant returns to scale.
B. constant returns to scale.
C. Decreasing returns to scale
D. increasing returns to scale.
E.increasing returns to scale
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Exercise
Medical Testing Labs, Inc., provides routine testing servicesfor blood banks in the Los Angeles area. Tests are supervisedby skilled technicians using equipment produced by twoleading competitors in the medical equipment industry.Records for the current year show an average of 27 tests per
hour being performed on the Testlogic-1 and 48 tests perhour on a new machine, the Accutest-3. The Testlogic-1 isleased for $18,000 per month, and the Accutest-3 is leased at$32,000 per month. On average, each machine is operated 25eight-hour days per month.
a. Does Medical Testing Lab usage reflect an optimal mix oftesting equipment?
b. If tests are conducted at a price of $6 each while labor and allother costs are fixed, should the company lease moremachines?
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a) (27*25*8)/ 18000 = (48*25*8) / 32000 = 0.3
In both instances, the last dollar spent on each machineincreased output by the same 0.3 units, indicating anoptimal mix of testing machines.
b) For each machine hour, the relevant question isTestlogic-1
27 (25 8) $6 > $18,000 or $32,400 > $18,000.
Accutest-3
48 (25 8) $6 > $32,000 or $57,600 > $32,000.In both cases, each machine returns more than its
marginal cost (price) of employment, and expansionwould be profitable.
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The marginal product of labor for international
trading is given by the equation
MPL = 10K0.5/L0.5Currently the firm is using 100 units of capital and
121 units of labor. The capital stock is constant but
the labor can be varied. If the price of labor is 10/-
and price of output is Rs. 2/- per unit, is the firm
operating efficiently in the short run? If not,
determine the optimal rate of labor input.
Exercise
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Answer: not optimally, L = 400
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The production function is : Q = 20K0.5L0.5
With marginal product functions
MPK= 10L0.5/K0.5MPL= 10K
0.5/L0.5
If the price of capital is Rs. 5/- and price of labor is Rs.4/- per unit, determine the expansion path for the firm.
The firm currently is producing 200 units of output perperiod using input rates of L = 4 and K =25. is this anefficient input combination? Why or why not? If not,determine the efficient input combination for producingan output rate of 200.
If the price of labor increases from Rs 4 to Rs 8 per unit,determine the efficient input combination for an outputrate of 200. What is the capital labor ratio now?
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Answer: K= 0.8L,
L= 11.18 and K= 8.94,
L = 7.905 and K = 12.65
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Suppose the price of one unit of labor is $10
and price of one unit of capital is $2.50. Use this information to determine the isocost
equations corresponding to a total cost of $200 and$500.
Plot these two isocost lines on a graph
If the price of labor falls from $10 per unit to $8 perunit, determine the new $500 isocost line and plot it
on the same diagram used in part (b) Answer: K = 2004L and K = 80- 4L, K =
2003.2L
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4. Given the production function Q = 30K0.7L0.5
and input prices r = 20 and w = 30.
Determine an equation for the expansion path
What is the efficient input combination for an
output rate of Q = 200? For 500?
Answer: K = 2.1L, for 200: L = 3.15 and K = 6.62,for 500: L = 6.765 and K = 14.207
Th d t f t t t l tifi d bli
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The revenue dept. of a state govt. employs certified publicaccountants (CPAs) to audit corporate tax returns and
book keepers to audit individual returns. CPAs are paid
$31200 per yr, while the annual salary of a bookkeeper is$18200. Given the current staff of CPAs and bookkeepers,a study made by the depts economist shows that addingone year of a CPAs time to audit corporate returns resultsin an additional tax collection of $52000. In contrast, an
additional bookkeeper adds $41600 per year in additionaltax revenue. If the depts objective is to maximize tax revenue collected, is
the present mix of CPAs and bookkeepers optimal? Explain
If the present mix of CPAs and bookkeepers is not optimal,
explain what re-allocation should be made. That is, should thedepartment hire more CPAs and fewer bookkeepers or viceversa.
Answer: CPAs1.67 and bookkeepers2.29