THE THEORY AND ESTIMATION OF PRODUCTION 1
THE THEORY AND ESTIMATION OF PRODUCTION
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PRODUCTION FUNCTION
Production function: defines the relationship between inputs and the maximum amount that can be produced within a given period of time with a given level of technology
Q = f(X1, X2, ..., Xk)
Q = level of output
X1, X2, ..., Xk = inputs used in production
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PRODUCTION FUNCTION
Short-run production function: the maximum quantity of output that can be produced by a set of inputs Assumption: the amount of at least one of the
inputs used remains unchanged
Long-run production function: the maximum quantity of output that can be produced by a set of inputs Assumption: the firm is free to vary the amount
of all the inputs being used
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SHORT RUN VS. LONG RUN
The short run is defined as the period of time when the plant size is fixed.
The long run is defined as the time period necessary to change the plant size.
Duration of the long/short run depends on the production process…
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Plant size is fixed, labor is variable
Both Plant size and labor
are variable
SHORT RUN VS. LONG RUN
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Plant size is fixed, labor is variable
Short Run
To increase production firms increase Labor but can’t expand their plant
Short Run
Firms produce in the short run
SHORT RUN VS. LONG RUN
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Plant size is variable, labor is variable
Long Run
To increase production firms increase Labor and expand their plant.
Long Run
Firms plan in the long run
How can the plant size be
variable?Plant size is
variable in the ‘planning’
stage
SHORT-RUN ANALYSIS OF TOTAL,AVERAGE, AND MARGINAL PRODUCT
Alternative terms in reference to inputs ‘inputs’ ‘factors’ ‘factors of production’ ‘resources’
Alternative terms in reference to outputs ‘output’ ‘quantity’ (Q) ‘total product’ (TP) ‘product’
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THERE ARE THREE IMPORTANT WAYS TO MEASURE THE PRODUCTIVITY OF LABOR:
Total product (TP)Average product (AP)Marginal product (MP)
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TOTAL PRODUCT FUNCTION (TP)
Represents the relationship between the number of workers (L) and the TOTAL number of units of output produced (Q) holding all other factors of production (the plant size) constant.For a coffee shop, output would be
measured in “number of coffee cups a day”For a steel mill, output would be measured
in “tons of steel produced a day” 9
BUILDING A TOTAL PRODUCT GRAPH
The Total Product Curve must show that:
1. With more workers more output can be produced.
INCREASING FUNCTION.INCREASING FUNCTION.
Labor
To
tal
Pro
du
ct
Labor
To
tal
Pro
du
ct
Labor
To
tal
Pro
du
ct
Constant Slope
1 2 3 4 5
5
10
15
20
25
5
5
5
5
5
Constant
Number of Workers hired
Number of units of output produced
0
Output increases by the same amount for each worker
hired
Output increases by the same amount for each worker
hired
Increasing Slope
1 2 3 4 5
5
15
30
50
75
10
15
20
25
Increasing
ALL workers become more productive as
they concentrate on doing only one
task
ALL workers become more productive as
they concentrate on doing only one
task
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Output increases by increasing amounts for each worker
hired
Output increases by increasing amounts for each worker
hired
Decreasing Slope
1 2 3 4 5
25
75
60
45
705
10
15
20
Decreasing
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ALL workers become LESS
productive as the plant gets
crowded and equipment breaks
down often
ALL workers become LESS
productive as the plant gets
crowded and equipment breaks
down often
Output increases by decreasing
amounts for each worker hired
Output increases by decreasing
amounts for each worker hired
1 3 5 7 95
15
30
50
75
5
10
15
20
25
2 4 6 8 10
95
120125
110
510
15
20
Positive Increasing and Positive Decreasing SlopeIncreasing Decreasing
1 3 5 7 95
15
30
50
75
510
15
20
25
2 4 6 8 10
95
120125
110
510
15
20
Positive Increasing, Positive Decreasing and Negative Slope
-5-10
-15
11 12
ALL THREE FUNCTIONS ARE INCREASING….Q
As L increases, Q increase by the same amount
Constant Slope
L
Increasing Slope
As L increases, Q increase by increasing amounts
L
Q
Decreasing Slope
As L increases, Q increase by decreasing amounts
L
Q
Larger steps
Smaller steps
Same size steps
WHICH OF THESE THREE SHAPES BEST DESCRIBES WHAT IS COMMON TO MOST PRODUCTION PROCESSES?
In other words: Does each additional worker add the SAME? MORE? Or LESS to output that the previous worker?
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FOR MOST PRODUCTION PROCESSES
In the short run, the plant size is fixed. Adding more workers is favorable to
production at first, as specialization increases productivity.
Eventually, adding more and more workers to a FIXED PLANT size results in decreases in productivity due to “crowded conditions”: Workers will have to SHARE EXISTING
EQUIPMENTEquipment will break down more often.
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THE LAW OF DIMINISHING MARGINAL PRODUCT.
As more of a variable input (labor) is added to a fixed input (plant), additions to output eventually slow down.
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NEGATIVE MARGINAL PRODUCT
If more of the variable input (labor) continues to be added to a fixed input (plant), additions to output continue to decline until eventually output decreases
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CHOOSING THE SLOPE:
2. For most productions processes as we add more workers, additions to output increase at the beginning but eventually decrease (could become negative).
For this, we use a function with both increasing and decreasing steps.
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The most common production function has increasing slope at the
beginning. Eventually, slope decrease and slope may
become negative
MARGINAL PRODUCT (MP) The additional output that can be produced by adding one more worker while holding plant size constant.
MP = Q/LIs the slope of the Total Product
Function22
MP: SLOPE OF THE PRODUCTION FUNCTION
Q (units produced)
L (Workers hired)10
160 units TP(Q)
Slope = 30/1 = 30MP = 30
Rise Q
Run L
9
130 units
30 units
1
The 10th worker adds 30 units to production
MPMP
MP: SLOPE OF THE PRODUCTION FUNCTION
Q
L12
160 units TP
Slope = 30/3 = 10
MP = 10
Rise
Run
9
130 units30
3
Each one of these three
workers adds 10 units to
production
MPMP
MP INCREASES AND DECREASES WHILE TOTAL PRODUCT STILL RISING
1 2 3 4
8
20
2527
Q
1st 4th3rd2nd
MP = 8
MP = 12
MP = 5
MP = 2
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5thMP = -4
If more workers are added, MP turns NEGATIVE
8
12
52 -4
1 2 3 4
5
MP
5
TOTAL PRODUCT VS. MARGINAL PRODUCT
MP = 8
MP = 12
MP = 5
MP = 2
MP = -4 1 2 3 45
MP
1 2 3 4
8
20
2527
Q
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5
TP rises up to 4th worker
MP falls after to
2nd worker
MP becomes negative after
4th worker
TP falls after 4th worker
MP rises up to 2nd worker
L MP Q
0
1 5
2 10
3 15
4 20
5 25
6 30
7 35
8 40
9 45
10 50
11 55
12 60
L MP Q
0 0
1 60
2 115
3 165
4 210
5 250
6 285
7 315
8 340
9 360
10 375
11 385
12 390
In this table: you’re given the Marginal Product and
you must use it to calculate the Total Product.
In this table: you’re given the Total Product and you must use it to calculate the
Marginal Product.
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L MP Q L MP Q 0 0 01 5 5 1 60 602 10 15 2 55 1153 15 30 3 50 1654 20 50 4 45 2105 25 75 5 40 2506 30 105 6 35 2857 35 140 7 30 3158 40 180 8 25 3409 45 225 9 20 360
10 50 275 10 15 37511 55 330 11 10 38512 60 390 12 5 390
AVERAGE PRODUCT (AP)
Represents the amount of output produced by each worker on average.
Or Output per worker.
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Output per worker = 15
units
Slope of that ray= Q/L = AP
Q
L10
150 unitsTP
AP = 150/10 = 15
When 10 workers produce 150 units,
Rise
Run
Q
L
AP = Q/L
OUTPUT PER WORKER
If we draw a line (a ray) from the
origin to a point on the production function
OUTPUT PER WORKER: AVERAGE PRODUCT (AP)
AP = Q/L AP = SLOPE OF RAY FROM ORIGIN
Q L AP
5 5 1.00
20 10 2.00
30 12 2.50
70 16 4.38
80 20 4.00
82 23 3.57
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Q
L
70
TP
What happens to the slope as L increases?
What happens to the slope as L increases?
8280
30
20
5
5 1012 16 20 23
What happens to the AP as L
increases?
What happens to the AP as L
increases?
AP: INCREASES, REACHES A MAXIMUM AND DECREASES.
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AP
L16
AP Increases up to 16 workers
AP Decreases after L=16
70/16=4.38
L
Q L AP
5 5 1.00
20 10 2.00
30 12 2.50
70 16 4.38
80 20 4.00
82 23 3.57
THE RELATIONSHIP BETWEEN AP AND MP
If MP (70) > AP (60), then the Average Product increases.
If MP (50) < AP (60), then the AP will decrease.
If MP = AP, then the AP is not increasing or decreasing: it is at the maximum point.
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If your next grade is say 70 > your test average so far say 60, then your test Average increases.
If your next grade is say 50 < your test average so far say 60, then your test Average decreases.
If your next grade is 60 = your test average so far 60, then your test Average stays the same .
If the MP of the next worker is say 50 < per worker average so far say 60, then the per worker average (AP) decreases.
If the MP of the next worker is say 70 > per worker average so far say 60, then the per worker average (AP) increases.
If the MP of the next worker is say 60 = per worker average so far say 60, then the per worker average (AP) stays the same.
MP AND AP
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MP
AP
MP AP 10
5
8
AP of 8 workers = 35/8 = 4.44.4
Marginal product of 9th worker = 10
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Suppose that 8 workers produce a total of 35 units9 workers produce a total of 45 units
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AP of 9 workers = 45/9=5
AP incr
eases
MP
> A
P
MP AND AP
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MP
AP
MP AP
5.9
AP of 12 workers = 71/12 = 5.9
5.9
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Suppose that 12 workers produce a total of 71 units13 workers produce a total of 76.9 units
AP of 13 workers = 76.9/13 = 5.9
AP remains same
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AP = MP=5.95.9
MP = 5.9
RELATIONSHIP BETWEEN MP AND AP
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MP
AP
AP incr
eases
MP below AP
MP above AP AP decreases
MP APMP = AP, AP doesn’t
change and AP is max
70
60
LONG-RUN PRODUCTION FUNCTION
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In the long run, a firm has enough time to change the amount of all its inputs
The long run production process is described by the concept of returns to scale
Returns to scale = the resulting increase in total output as all inputs increase
LONG-RUN PRODUCTION FUNCTION
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If all inputs into the production process are doubled, three things can happen:
output can more than double‘increasing returns to scale’ (IRTS)
output can exactly double‘constant returns to scale’ (CRTS)
output can less than double‘decreasing returns to scale’ (DRTS)
LONG-RUN PRODUCTION FUNCTION
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One way to measure returns to scale is to use a coefficient of output elasticity:
if EQ > 1 then IRTS if EQ = 1 then CRTS if EQ < 1 then DRTS
inputsallinchangePercentage
QinchangePercentageQE
LONG-RUN PRODUCTION FUNCTION
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Graphically, the returns to scale concept can be illustrated using the following graphs
Q
X,Y
IRTSQ
X,Y
CRTSQ
X,Y
DRTS
ESTIMATION OF PRODUCTION FUNCTIONS
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Production function examples
•short run: one fixed factor, one variable factor
Q = f(L)K
•cubic: increasing marginal returns followed by decreasing marginal returns
Q = a + bL + cL2 – dL3
•quadratic: diminishing marginal returns but no Stage I
Q = a + bL - cL2
ESTIMATION OF PRODUCTION FUNCTIONS
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Production function examples
•Cobb-Douglas function: exponential for two inputs Q = aLbKc
if b + c > 1, IRTS if b + c = 1, CRTS if b + c < 1, DRTS
ESTIMATION OF PRODUCTION FUNCTIONS
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Statistical estimation of production functions
• inputs should be measured as ‘flow’ rather than ‘stock’ variables, which is not always possible
• usually, the most important input is labor
• most difficult input variable is capital
• must choose between time series and cross-sectional analysis
IMPORTANCE OF PRODUCTION FUNCTIONS IN MANAGERIAL DECISION MAKING
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Careful planning can help a firm to use its resources in a rational manner.
• Production levels do not depend on how much a company wants to produce, but on how much its customers want to buy.
• There must be careful planning regarding the amount of fixed inputs that will be used along with the variable ones.
IMPORTANCE OF PRODUCTION FUNCTIONS IN MANAGERIAL DECISION MAKING
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Capacity planning: planning the amount of fixed inputs that will be used along with the variable inputs
Good capacity planning requires:
• accurate forecasts of demand• effective communication between the production
and marketing functions
IMPORTANCE OF PRODUCTION FUNCTIONS IN MANAGERIAL DECISION MAKING
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• The intensity of current global competition often requires managers to go beyond these simple production function curves.
• Being competitive in production today mandates that today’s managers also understand the importance of speed, flexibility, and what is commonly called “lean manufacturing”.
THANK YOU
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