Prod functionsp06

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


5

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


6

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’

7

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


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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)


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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


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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


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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


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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.


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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


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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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Prod functionsp06