Production

Outline:

•Introduction to the production function

•A production function for auto parts

•Optimal input use

•Economies of scale

•Least-cost production

The production function

•Production is the process of transforming inputs into semi-finished articles (e.g., camshafts and windshields) and finished goods (e.g., sedans and passenger trucks).

•The production function indicates that maximum level of output the firm can produce for any combination of inputs.

General description of a production function

Let:

Q = F (M, L, K) [1]

Where Q is the quantity of output produced per unit of time (measured in units, tons, bushels, square yards, etc.), M is quantity of materials used in production, L is the quantity of labor employed, and K is the quantity of capital employed in production.

Technical efficiency

The production function indicates the maximum output that can be obtained from a

given combination of inputs—that is, we assume the firm is

technically efficient.

A production function for auto parts

Consider a multi-product firm that supplies parts to major U.S. auto manufacturers. Its production function is given by Let

Q = F(L, K)

Where Q is the quantity of specialty parts produced per day, L is the number of workers employed per day, and K is plant size (measured in thousands of square feet).

Number of Plant Size (000s)

Workers 10 20 30 4010 93 120 145 16520 135 190 235 26430 180 255 300 33740 230 315 365 41050 263 360 425 46060 293 395 478 51070 321 430 520 55580 346 460 552 60090 368 485 580 645

100 388 508 605 680

This table [1] shows the quantity of output that can be obtained from various combinations of plant size and labor

The short run

•Inputs that cannot be varied in the short run are called fixed inputs.

•Inputs that can vary are called (not surprisingly) variable inputs

The short run refers to the period of time in which one or more of

the firm’s inputs is fixed—that is, cannot be

varied

The long runThe long run is the period of time sufficiently long to allow the firm to vary all inputs—e.g., plant size,

number of trucks, or number of apple trees.

Marginal product

•Marginal product is the additional (or extra) output resulting from the employment of one more unit of a variable input , holding all other inputs constant.

•In our example, the marginal product of labor (MPL) is the extra output of auto parts realized by employing one additional worker, holding plant size constant

Number of Total MarginalWorkers Product Product

10 9320 135 4.230 180 4.540 230 550 263 3.360 293 370 321 2.880 346 2.590 368 2.2

100 388 2.0110 400 1.2120 403 0.3130 391 -1.2140 380 -1.1

Production of specialty parts, assuming a plant size of 10,000 square feet

Law of diminishing returns As units of a variable input are added (with all other inputs held constant), a point is reached where additional units will add successively decreasing increments to total output—that is, marginal product will begin to decline.

Notice that, after 40 workers are employed, marginal product begins to decline

500

400

300

200

100

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

Total Output

20,000-square-foot plant

10,000-square-foot plant

Number of Workers

The total product of labor

The marginal product of labor when plant size is 10,000 square feet

5.0

4.0

3.0

2.0

1.0

010 20 30 40 50 60 70 80 90 100 110 130 140

Marginal Product

Number of Workers

–1.0

–2.0

120

Optimal use of an input

By hiring an additional unit of labor, the firm is adding to its costs—but it is also adding to its output and thus revenues.

Marginal revenue product of labor (MRPL)

The marginal revenue product of labor (MRPL) is given by

MRPL = (MR)(MPL) [6.2]

Where MR marginal revenue—that is, the additional (extra) revenue realized by selling one more unit.

Example: If MPL is 5 units, and the firm can sell additional units for $6 each, then:

MRPL = (MR)(MPL) = (5)($6) = $30

Marginal cost of labor (MCL)

What additional cost does the firm incur (wages, benefits,

payroll taxes, etc.) by hiring one more worker?

-maximizing rule of thumbThe firm should employ additional units of the variable input (labor) up

to the point where MRPL = MCL1

1In terms of calculus, we have:

MRPL = (MR)(MPL) = (dR/dQ)(dQ/dL) and

MCL = dC/dL

Example

Example:

• The firm has estimated that the cost of hiring an additional worker is equal to $160 per day, that is, MCL = PL = $160.

•Assume the firm can sell all the parts it wants at a price of $40. Hence, MR = $40

•Thus the MRPL = (MR)(MPL) = ($40)(MPL)

Number of Total Marginal Marginal MarginalWorkers Product Product Revenue Product Cost

10 93 16020 135 4.2 168 16030 180 4.5 180 16040 230 5 200 16050 263 3.3 132 16060 293 3 120 16070 321 2.8 112 16080 346 2.5 100 16090 368 2.2 88 160

100 388 2.0 80 160110 400 1.2 48 160120 403 0.3 12 160130 391 -1.2 -48 160140 380 -1.1 -44 160

ProblemLet the production function be given by:

Q = 120L – L2

The cost function is given by

C = 58 + 30L

The firm can sell an unlimited amount of output at a price equal to $3.75 per unit

1. How many workers should the firm hire?

2. How many units should the firm produce?

Production in the long run

•The scale of a firm’s operation denotes the levels of all the firm’s inputs.

•A change in scale refers to a given percentage change in all the firm’s inputs—e.g., labor, materials, and capital.

•If we say “the scale of production has increased by 15 percent,” we mean the firm has increased its employment of all inputs by 15 percent.

Returns to scale

Returns to scale measure the percentage

change in output resulting from a given percentage change in

inputs (or scale)

3 cases

1. Constant returns to scale: 10 percent increase in all inputs results in a 10 percent increase in output.

2. Increasing returns to scale: 10 percent increase in all inputs results in a more than 10 percent increase in output.

3. Decreasing returns to scale: 10 percent increase in all inputs results in a less than 10 percent increase in output.

Sources of increasing returns1. Specialization of plant and equipment

Example:Large scale production in furniture manufacturing allows for application of specialized equipment in metal fabrication, painting, upholstery, and materials handling.

2. Economies of increased dimensionsExample: Doubling the circumference of pipeline results in a fourfold increase in cross sectional area, and hence more than doubling of capacity, measured in gallons per day.

3. Economies of massed reserves.Example: A factory with one stamping machine needs to have spare 100 parts in inventory to be prepared for breakdown—does a factory with 20 machines need to have 2,000 spare parts on hand?

Economies of increased dimensions

r

h

rhrhSA 22 2

hrV 2

Effect of a 1 inch change in vessel radius

Surface Area and Volume

PI r (in.) h (in.) S.A. (sq. in.) V (cu. in.)3.1416 6 10 4146.902 1130.9733.1416 7 10 4838.053 1539.380

Change in S.A (%) Change in V. (%)16.6667 36.111

Intermodal Freight Containers

The shift from the 20 foot to the 40 foot freight container hasmade shipping goodsmore economical

See link

Fixed and Sunk Costs

•Fixed costs (FC) are elements of cost that do not vary with the level of output.

Examples: Interest payments on bonded indebtedness, fire insurance premiums, salaries and benefits of managerial staff.

•Sunk costs are costs already incurred and hence non-recoverable.

Examples: Research & development costs, advertising costs, cost of specialized equipment.

DefinitionsVariable cost (VC) is the sum of the firm’s expenditure for variable inputs such as hourly employees, raw materials or semi-finished articles, or utilities.

Average total cost (SAC) is total cost divided by the quantity of output.

Average variable cost (AVC) is variable cost divided by the quantity of output.

Marginal cost (SMC) is the addition to total cost attributable to the last unit produced

Annual Output Total Cost Fixed Cost Variable Cost(Thousands of Repairs) ($000s) ($000s) (000s)

0 270.0 270 0.05 427.5 270 157.510 600.0 270 330.015 787.5 270 517.520 990.0 270 720.025 1207.5 270 937.530 1440.0 270 1170.035 1687.5 270 1417.540 1950.0 270 1680.045 2227.5 270 1957.550 2520.0 270 2250.055 2827.5 270 2557.560 3150.0 270 2880.0

Firm’s Costs in the Short Run

3,000

0 60O utput (Thousands of Units)

Tota l Cost (Thousands of Dollars)

555045403530252015105

2,000

1,000

Cost function

Annual Output Total Cost Ave. Cost Marginal Cost(Thousands of Repairs) ($000s) ($000s) ($000s)

0 270.05 427.5 85.5 31.510 600.0 60.0 34.515 787.5 52.5 37.520 990.0 49.5 40.525 1207.5 48.3 43.530 1440.0 48.0 46.535 1687.5 48.2 49.540 1950.0 48.8 52.545 2227.5 49.5 55.550 2520.0 50.4 58.555 2827.5 51.4 61.560 3150.0 52.5 64.5

64

0 60O utput (Thousands of Units)

Cost/Unit (Thousands o f Do lla rs)

44

48

52

56

555045403530252015105

SM C

SAC

60

Relationship between Average and Marginal

When average cost is falling, marginal cost lies everywhere below average cost.

When average cost is rising, marginal cost lies everywhere above average cost.

When average cost is at its minimum, marginal cost cost is equal to average cost.

If your most recent (marginal) grades are higher than your GPA at the start of the term,

your GPA will rise

What explains rising (short-run) marginal cost?

If labor is the only variable input then marginal cost can be expressed by:

L

L

MP

PSMC [7.1]

Recall that the marginal product of labor will begin to fall at some point due to the law of diminishing returns.

0.0

50.0

100.0

150.0

200.0

250.0

300.0

1 11 21 31 41 51

Output (Q)

Fixe

d C

ost P

er U

nit

Behavior of Average Fixed Cost

As output increases, fixed cost can be spread more thinly

Production costs is the long run

•In the long run there are no fixed inputs; hence all costs are “variable.”

•The long run average cost curve shows the minimum average cost achievable at each level of output in the long run—that is, when all inputs are variable.

Constant Returns to Scale

$5

0O utput (Thousands o f Units )

Long -R un A ve rage C ost

4

21614410872

S A C 1

(9 ,000 -square -foo t p lant)

-S A C 2

(18 ,000-squarefoo t p lant) (

S A C 3

27 ,000-squa re -foo t p lant)

S M C 1 S M C 2 S M C 3

LA C = LM C

The U-Shaped Long Run Average Cost Function

Output

Long-Run Average Cost

SAC1

SMC2

Qmin

SMC1 SMC3

SAC2 SAC3

LMC

LMC

LAC

Increasing returns Decreasing returns

Notice on the previous slide that up to a scale of QMIN, the firm experiences decreasing

(long run) unit cost. Economies of scale are exhausted at the

point

Minimum Efficient Scale (QMES)

QMES is the minimum scale of operation at which long unit

production costs can be minimized.

LAC

Demand

QMES is large relative to he “size of the market.”

Q

Cost

per

unit

0 1000 2000

To produce on an efficient scale, you must supply 50% of the product demanded at a price equal to minimum unit cost

How large do you have to be to minimize unit costs?1

Not very large (as a percent of U.S. consumption) : Bricks, flour milling, machine tools, cement, glass containers, cigarettes, shoes, bread baking.

Fairly large (as a percent of U.S. consumption): Synthetic fibers, passenger cars, household refrigerators and freezers, commercial aircraft.

Very large (as a percent of U.S. consumption): Turbine generators, diesel engines, electric motors, mainframe computers.

1F.M. Scherer and D. Ross. Industrial Market Structure and Performance, 3rd edition, 1990, pp. 115-116.

Output

Long-Run Average Cost

Qmin(a)

Output

Long-Run Average Cost

Qmin(b)

Output

Long-Run Average Cost

(c)

Local telephone service, electricity distribution, and cable TV distribution are well represented by this cost function.