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The Production FunctionChapter 9: Production and Cost Analysis I
04
/15
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1
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…
2
Plant size is fixed, labor is variable
Both Plant size and labor
are variable
Short Run vs. Long Run
3
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
4
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
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 (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”
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6
Building a Total Product Graph
The Total Product Curve must show that:
1. With more workers more output can be produced.
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INCREASING FUNCTION.INCREASING FUNCTION.
Labor
To
tal
Pro
du
ct
Labor
To
tal
Pro
du
ct
Labor
To
tal
Pro
du
ct
Marginal ProductMarginal = additional
Marginal Product is the additional output produced by the last worker hired.
04
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8
If TP has a Constant Slope
1 2 3 4 5
5
10
15
20
25
+5
+5
+5
+5
+5
Constant
Number of Workers hired
Units produced
0
Output increases by
the same amount for each worker
hired
Output increases by
the same amount for each worker
hiredThis is
an in
crea
sing fu
nctio
n with
a c
onstan
t slo
pe
If TP has a Constant Slope
1 2 3 4 5
Constant
Worker #
Marginal Product
Output increases by
the same amount for each worker
hired
Output increases by
the same amount for each worker
hired
5+5 +5 +5 +5 +5
Marginal Product
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
5
Output increases by increasing amounts for each worker
hired
Output increases by increasing amounts for each worker
hired
This
is a
n in
crea
sing
func
tion
with
an
incr
easi
ng s
lope
Increasing Slope
1 2 3 4 5
5101520
25
1015
2025
Increasing
5
Output increases by increasing amounts for each worker
hired
Output increases by increasing amounts for each worker
hired
Marginal Product
Marginal Product
Worker #
Decreasing Slope
1 2 3 4 5
25
75
60
45
705
10
15
20
Decreasing
25
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
This is
an in
crea
sing fu
nctio
n with
an D
ecre
asin
g slo
pe
1 2 3 4 5
510152025
1015
2025
Decreasing
5
Output increases by decreasing amounts for each worker
hired
Output increases by decreasing amounts for each worker
hired
Marginal Product
Marginal Product
Worker #
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 the Marginal Product increase, decrease or remains the same as workers are hired?
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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.
18
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 decrease
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19
Choosing the best shape for the production function:
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 (eventually negative) MP
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20
The most common production function has increasing slope at the
beginning. Eventually, slope decrease and slope may
become negative
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
Marginal Product (MP) The additional output that can be produced by adding more workers to a constant size plant.
MP = Q/LIs the slope of the Total Product
Function
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23
MP: Slope of the Production FunctionQ (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
23
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
23
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
Diminishing Returns to Labor set in after worker 2
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.
29
L Q MP
0 0 1 10 2 26 3 36 4 44 5 50 6 54 7 56 8 55 9 53
10 50 11 46
L Q MP0 0 1 10 102 26 163 36 104 44 85 50 66 54 47 56 28 55 -19 53 -2
10 50 -311 46 -4
30
L Q MP0 1 102 153 204 185 146 107 58 29 0
10 -311 -5
L Q MP0 1 10 102 25 153 45 204 63 185 77 146 87 107 92 58 94 29 94 0
10 91 -311 86 -5
31
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)
Output per worker
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32
AP = Total Product / LaborAP = Total Product / Labor
AP = Q/LAP = Q/L
Output per worker = 15
unitsQ
L10
150 unitsTP
AP = 150/10 = 15
Output per worker
Draw a line (a ray) from the origin to any point on
the production function
Draw a line (a ray) from the origin to any point on
the production function
Output per worker: Average Product (AP)
Slope of that ray= Q/L = AP
If 10 workers produce 150 units,
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
34
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.
35
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.
36
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.
THE AP AND MP…
37
Slope of ray is max
Changes concavity
MP
AP
TP
MP, APL
L
AP is max
MP is maxDRT set in
DRT set in
TP is max
MP is zero
MP and AP
38
MP
AP
MP AP 10
5
8
AP of 8 workers = 35/8 = 4.44.4
Marginal product of 9th worker = 10
9
Suppose that 8 workers produce a total of 35 units9 workers produce a total of 45 units
9
AP of 9 workers = 45/9=5
AP incr
eases
MP
> A
P
MP and AP0
4/1
5/2
3
39
MP
AP
MP AP
5.9
AP of 12 workers = 71/12 = 5.9
5.9
13
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
12
AP = MP=5.95.9
MP = 5.9
Relationship between MP and AP0
4/1
5/2
3
40
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
MARGINAL REVENUE PRODUCT
MRP = Revenue generated by last worker hiredMP = Units added to total product by last worker hired. MRP = MP * Price
The firm should hire all workers for whom the revenue each generate exceeds his wage.
The firm should hire all workers for whom the MRP > wage.
41
HOW MANY WORKERS SHOULD BE HIRED?
42
L TP 0 01 152 273 374 445 476 497 488 45
L TP MP0 0 1 15 152 27 123 37 104 44 75 47 36 49 27 48 -18 45 -3
L TP MPMRP
0 0 MP*Price
1 15 15 4502 27 12 3603 37 10 3004 44 7 2105 47 3 906 49 2 607 48 -1 -308 45 -3 -90
Worker one adds 15 units to output which bring $450 dollars in additional revenue.
This additional revenue is larger than his salary
($200) so the firm should hire this worker
Worker two adds 12 units to output which bring $360 dollars in additional revenue.
This additional revenue is larger than his salary
($200) so the firm should hire this worker
Worker three adds 10 units to output which bring $300 dollars in additional revenue.
This additional revenue is larger than his salary
($200) so the firm should hire this worker
Worker four adds 7 units to output which bring
$210 dollars in additional revenue.
This additional revenue is larger than his salary
($200) so the firm should hire this worker
Worker 5 adds 3 units to output which bring
$90 dollars in additional revenue.
This additional revenue
is smaller than his salary ($200) so the firm
should NOT hire this worker
When the wage is $200: Demand for workers is 4
Wage per worker per day = $200 Price per unit = $30
BUILDING LABOR DEMAND LINE
43
L TP MPMRP
0 0 MP*Price
1 15 15 4502 27 12 3603 37 10 3004 44 7 2105 47 3 906 49 2 607 48 -1 -308 45 -3 -90
If wage is:The firm will
hire _ workers
450 1360 2300 3210 490 560 650 ?40 ?
The firm should hire all workers for whom the MRP greater than or equal to the wage.
THE OPTIMAL USE OF AN INPUT
44
L TP MPMRP
0 0 MP*Price
1 15 15 4502 27 12 3603 37 10 3004 44 7 2105 47 3 906 49 2 607 48 -1 -308 45 -3 -90
Once diminishing returns to labor set in
the MP decreases
When the MP decreases, the MRP
also decreases
The firm should hire more workers as long as
the MRP > wage
We know the firm has hired the optimum
number of workers when the MRP = wage
Rule: Increase use of an input until
MPR of that input = Price of the input
45
L Q MP0 1 10 102 25 153 45 204 63 185 77 146 87 107 92 58 94 29 94 0
10 91 -311 86 -5
Wage Labor Demand51030 60 90 120 150 180 190 220
Price = $10L Q MP MRP0 1 10 102 25 153 45 204 63 185 77 146 87 107 92 58 94 29 94 0
10 91 -311 86 -5
L Q MP MRP
1 45 20 200
2 63 18 180
5 77 14 140
4 87 10 100
5 92 5 50
6 94 2 20
7 94 0 0
8 91 -3 -30
9 86 -5 -50
CONSIDER A SMALL SANDWICH SHOP…0
4/1
5/2
3
46
L Q MP AP0 01 10 10 10.02 25 15 12.5
2.5 31.3 12.5 12.53 35 10 11.74 40 5 10.05 42 2 8.46 42 0 7.07 35 -7 5.08 25 -10 3.19 10 -15 1.1
# sandwiches# workers
MP= previous AP
AP doesn’t change
In this table: you’re given the Marginal Product and
you must use it to calculate the Total Product.
L MP Q AP
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 AP
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
Table 1 Table 2
L Q MP AP
0
10 5
20 25
30 70
40 110
50 135
60 153
70 118
80 38
L MP Q AP
0
10 5
20 20
30 45
40 40
50 25
60 18
70 -35
80 -80
Questions to practice for the test
Here you have the Total Product Q and
you must calculate the MP and AP Here you have
the Marginal Product MP and
you must calculate the Total Product and AP
Table 1 Table 2
L Q MP AP
0
10 5
20 25
30 70
40 110
50 135
60 153
70 118
80 38
L MP Q AP
0
10 5
20 20
30 45
40 40
50 25
60 18
70 -35
80 -80
Questions to practice for the test
Table 1 Table 2
L Q MP AP
0
1 5
2 25
3 70
4 110
5 135
6 153
7 118
8 38
L MP Q AP
0
1 5
2 20
3 45
4 40
5 25
6 18
7 -35
8 -80
Questions to practice for the test
L Q Q L MP
0 0
10 5 5 10 0.5
20 25 20 10 2
30 70 45 10 4.5
40 110 40 10 4
50 135 25 10 2.5
60 153 18 10 1.8
70 118 -35 10 -3.5
80 38 -80 10 -8
L Q Q L MP MP *10 Q
0 0 0
10 5 5 10 0.5 5 5
20 25 20 10 2 20 25
30 70 45 10 4.5 45 70
40 110 40 10 4 40 110
50 135 25 10 2.5 25 135
60 153 18 10 1.8 18 153
70 118 -35 10 -3.5 -35 118
80 38 -80 10 -8 -80 38
For each table in the next slides answer the following questions:1. What is the shape of the Total Product Curve? Should be
able to draw the total product curve.2. What is the shape of the Marginal Product Curve? Should
be able to draw the Marginal Product Curve.3. What is the shape of the Average Product Curve? Should be
able to draw the Average Product Curve.4. With which worker(s) do we realize
increasing/decreasing/negative marginal productivity? How do you know?
5. Would you employ the 6th worker? Why yes/why not?6. How are the marginal product and the average product
related?
Questions to practice for the test
Fill in the TP and AP Should be able to draw these graphs.
L MP TP(Q) AP
0
1 5
2 5
3 5
4 5
5 5
6 5
7 5
8 5
9 5
10 5
11 5
12 5
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Fill in the TP and AP Should be able to draw these graphs.
L MP TP (Q) AP
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
04
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Fill in the TP and AP Should be able to draw these graphs.
L MP TP (Q) AP
0
1 60
2 55
3 50
4 45
5 40
6 35
7 30
8 25
9 20
10 15
11 10
12 5
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56
Fill in the TP and AP Should be able to draw these graphs.
L MP TP(Q) AP
0
1 5
2 10
3 15
4 20
5 17
6 15
7 13
8 12
9 10
10 8
11 6
12 5
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57
A B
D E
C
F
MP=AP
APAP
AP
AP
AP
G
AP
MP
MP
MPMP
MP
MP
A B
D E
C
I II
IV
III
MPMP
MP MP MP
V
1 3 5 7 9
5
15
30
50
75
24 6 8 10
95
120
125
110
11 12