Prod Function

7/30/2019 Prod Function

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THE PRODUCTION FUNCTIONChapter 9: Production and Cost Analysis I

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A PLANT

Is the factory (space) whereproduction takes place and

All the equipment necessary toproduce

The plant includes everything neededto produce, except workers and raw

materials.

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SHORT RUNVS. LONG RUN

The short run is defined asthe period of time when theplant size is fixed.

The long run is defined asthe time period necessary tochange the plant size.

Duration of the long/shortrun depends on theproduction process

3

Plant size isfixed, laboris variable

Both Plantsize and laborare variable


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Plant size isfixed, laboris variable

Short Run

To increaseproduction firmsincrease Labor

but cant expandtheir plant

Short Run

Firms produce in the short run


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5

Plant size isvariable, labor isvariable

Long Run

To increaseproduction firmsincrease Labor

and expand theirplant.

Long Run

Firms plan in the long run

How canthe plantsize be

variable?


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THEREARETHREEIMPORTANTWAYSTO

MEASURETHEPRODUCTIVITYOFLABOR:

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 ofoutput

produced (Q) holding all other factorsof 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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BUILDINGA TOTAL PRODUCT GRAPH

The Total Product Curve must show that:

1. The more workers are hired the more

output would be produced.

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INCREASING FUNCTION.


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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 ofoutput produced

0


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Increasing Slope

1 2 3 4 5

5

15

30

50

75

10

15

20

25

Increasing

ALL workers become

more productive asthey concentrate ondoing only one task

5


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Decreasing Slope

1 2 3 4 5

25

75

60

45

705

1015

20

Decreasing

25

ALL workers becomeLESS productive as

the plant getscrowded and

equipment breaksdown often


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1 3 5 7 9

5

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


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ALL THREE FUNCTIONS ARE INCREASING.Q

As L increases, Q increases by

the same amount

Constant Slope

L

Increasing Slope


increasing amounts

L

Q

Decreasing Slope


decreasing amounts

L

Q

Larger steps

Smaller steps

Same size steps


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WHICHOFTHESETHREESHAPES

BESTDESCRIBESWHATISCOMMONTOMOSTPRODUCTIONPROCESSES?

In other words: Does each additional worker add theSAME? MORE? Or LESS to output that the previousworker?

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FORMOSTPRODUCTIONPROCESSES

In the short run, the plant size is fixed.

Adding more workers is favorable toproduction at first, as specialization

increases productivity.Eventually, adding more and more workers

to a FIXED PLANTsize results indecreasesin productivity due to crowded

conditions: Workers will have to SHARE EXISTING

EQUIPMENT

Equipment will break down more often.

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THE LAWOFDIMINISHING MARGINAL

PRODUCT.

As more of a variable input (labor) isadded to a fixed input (plant), additionsto output eventually slow down.

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NEGATIVE MARGINAL PRODUCT

If more of the variable input (labor)continues to be added to a fixedinput (plant), additions to outputcontinue to decline until eventually

output decreases

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CHOOSINGTHESLOPE:

2. For most productions processes as we addmore workers, additions to output increaseat the beginning but eventually decrease

(could become negative).

For this, we combine the graph with increasingsteps (slope) and the graph with decreasing

steps

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The most common productionfunction has an increasing slope

at the beginning, and eventually

decreasing slope which may

become negative


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MARGINAL PRODUCT (MP)The additional output that can be

produced by adding one more

worker while holding plant sizeconstant.

MP = Q/ L

Is the slope of the Total ProductFunction

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MP: SLOPEOFTHE PRODUCTION FUNCTION

Q (unitsproduced)

L (Workers hired)10

160 units TP(Q)

Slope = 30/1 = 30MP = 30

Rise Q

Run L

9

130 units

30 units

1

The 10thworker adds30 units toproduction


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MP: SLOPEOFTHE PRODUCTIONFUNCTION

Q

L12

160 units TP

Slope = 30/3 = 10MP = 10

Rise

Run

9

130 units

30

3

Each one ofthese three

workers adds 10units to

production


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MP INCREASES AND DECREASES WHILE

TOTAL PRODUCT STILL RISING

1 2 3 4

8

20

2527

1st 4th3rd2nd

MP = 8

MP = 12

MP = 5

MP = 2

23

5th MP = -4

If more workers are added, MP turns NEGATIVE

8

12

5

2 -4

1 2 3 4

5

MP

5


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TOTAL PRODUCTVS. MARGINAL PRODUCT

MP = 8

MP = 12

MP = 5

MP = 2

MP = -4 1 2 3 45

MP

1 2 3 4

8

20

2527

23

5


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TOTAL PRODUCTVS. MARGINAL PRODUCT

MP = 8

MP = 12

MP = 5

MP = 2

MP = -4 1 2 3 45

MP

1 2 3 4

8

20

2527

23

5

TP rises up to

4th worker

MP fallsafter to

2ndworker

MP becomes

negative after

4th worker

TP falls after4thworker

MP rises up

to 2ndworker


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L MP Q01 52 103 154 205 256 307 358 409 45

10 5011 5512 60

L MP Q0 01 602 1153 1654 2105 2506 2857 3158 3409 36010 37511 38512 390

In this table: youre given

the Marginal Product and

you must use it to calculate

the Total Product.


the Total Product and you

must use it to calculate the

Marginal Product.


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AVERAGE PRODUCT (AP)

Represents the amount of

output produced by eachworker on average.

Or Output per worker.

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Q

L10

150 units

TP

AP = 150/10 = 15

When 10 workersproduce 150 units,

Rise

Run

Output per

worker = 15units

Q

L

AP = Q/L

OUTPUTPERWORKER

Slope of that

ray= Q/L =AP

If we draw aline (a ray)from the

origin to apoint on theproduction

function


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AP = Q/L

AP = SLOPEOFRAYFROMORIGIN

Q L AP5 5 1.0020102.0030122.5070164.3880

20

4.00

82233.57

29

Q

L

70

TP

What happensto the slope as

L increases?

8280

30

20

5

5 10 12 16 20 23

What happensto the AP as L

increases?


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AP: INCREASES, REACHESAMAXIMUMAND

DECREASES.

30

AP

L16

AP Increases up

to 16 workers

AP Decreases

after L=16

70/16=4.38

L

Q L AP5 5 1.0020102.0030122.5070164.3880

20

4.00

82233.57


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THE RELATIONSHIPBETWEENAPANDMP

If MP (70) > AP (60), then the AverageProduct 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 testaverage 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 testaverage 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, thenthe per worker average (AP) increases.

If the MP of the next worker is say 60 = perworker average so far say 60, then the per

worker average (AP) stays the same.


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MPAND AP

33

MP

AP

MP AP 10

5

8 AP of 8 workers = 35/8 = 4.4

4.4

Marginal product of 9th worker = 10

9

Suppose that 8 workers produce a total of 35 units

So when the 9th worker is hired 9 workers produce a total of 45 units

9 AP of 9 workers = 45/9=5


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MPAND AP

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MP

AP

MP AP

5.9

AP of 12 workers = 71/12 = 5.9

5.9

Suppose that MP = AP: Marginal product of 13th worker = 5.9

13

Suppose that 12 workers produce a total of 71 units

So when the 13th worker is hired 13 workers produce a total of 76.9 units

AP of 13 workers = 76.9/13 = 5.9

AP remains same

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MP=5.95.9


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RELATIONSHIPBETWEEN MPAND AP 12

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MP

APMP below AP

MP above AP

MP AP MP = AP, AP doesntchange and

AP is max

70

6060

70


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CONSIDER A SMALL SANDWICH SHOP

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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 8.4

6 42 0 7.07 35 -7 5.08 25 -10 3.19 10 -15 1.1

# sandwiches# workers

MP= previous AP

AP doesnt

change

L MP Q AP


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the Marginal Product and

you must use it to calculate

the Total Product.

L MP Q AP0

1 5

2 10

3 15

4 205 25

6 30

7 35

8 40

9 45

10 50

11 55

12 60

L MP Q AP0 0

1 60

2 115

3 165

4 210

5 2506 285

7 315

8 340

9 360

10 375

11 385

12 390

Q i i f h


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Table 1 Table 2

L Q MP AP0

10 520 2530

70

40 11050 13560

153

70 11880 38

L MP Q AP0

10 520 2030

45

40 4050 2560

18

70 -3580 -80

Questions to practice for the test

Here you

have the Total

Product Q and

you mustcalculate the

MP and AP Here you have

the Marginal

Product MP and

you must

calculate the TotalProduct and AP

Q ti t ti f th t t


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Table 1 Table 2

L Q MP AP010 520 2530 7040 11050 13560

153

70 11880 38

L MP Q AP010 520 2030 4540 4050 2560

18

70 -3580 -80


Q ti t ti f th t t


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Table 1 Table 2

L Q MP AP01 52 253 704 1105 1356

153

7 1188 38

L MP Q AP01 52 203 454 405 256

18

7 -358 -80



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L Q DQ DL MP0 010 5 5 10 0.520 25 20 10 230 70 45 10 4.540 110 40 10 450 135 25 10 2.560 153 18 10 1.870 118 -35 10 -3.580 38 -80 10 -8


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L Q DQ DL MP MP *10 Q0 0 010 5 5 10 0.5 5 520 25 20 10 2 20 2530 70 45 10 4.5 45 7040 110 40 10 4 40 11050 135 25 10 2.5 25 13560 153 18 10 1.8 18 15370 118 -35 10 -3.5 -35 11880 38 -80 10 -8 -80 38


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For each table in the next slides answer the following

questions:

1. What is the shape of the Total Product Curve? Should beable 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 beable 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?



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FILLINTHE TPAND AP SHOULDBEABLE

TODRAWTHESEGRAPHS.

L MP TP(Q) AP01 52 53 54 55 56 57 58 59 510 511 512 5

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TODRAWTHESEGRAPHS.

L MP TP (Q) AP01 52 103 154 205 256 307 358 409 4510 5011 5512 60

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TODRAWTHESEGRAPHS.

L MP TP (Q) AP01 602 553 504 455 406 357 308 259 2010 1511 1012 5

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TODRAWTHESEGRAPHS.

L MP TP(Q) AP01 52 103 154 205 176 157 138 129 1010 811 612 5

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A B C


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A B

D E

C

F

MP=AP

APAP

AP

AP

AP

G

AP

MP

MP

MPMP

MP

MP


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A B

D E

C

I II IIIMP


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I II

IV

III

MPMP

MP MPMP

V

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Prod Function