Prod functions

The Production FunctionChapter 9: Production and Cost Analysis I

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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…

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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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.

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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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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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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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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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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    

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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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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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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