Transcript Production

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)
[6.1]
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).
This table [6.1] shows the quantity of
output that can be obtained from various
combinations of plant size and labor
Number of
Workers
10
20
30
40
50
60
70
80
90
100
Plant
10
93
135
180
230
263
293
321
346
368
388
Size
20
120
190
255
315
360
395
430
460
485
508
(000s)
30
145
235
300
365
425
478
520
552
580
605
40
165
264
337
410
460
510
555
600
645
680
The short run
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
•Inputs that cannot be varied in
the short run are called fixed
inputs.
•Inputs that can vary are called
(not surprisingly) variable
inputs
The 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.
The long run
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
Production of specialty parts,
assuming a plant size of
10,000 square feet
Number of
Workers
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Total
Marginal
Product Product
93
135
4.2
180
4.5
230
5
263
3.3
293
3
321
2.8
346
2.5
368
2.2
388
2.0
400
1.2
403
0.3
391
-1.2
380
-1.1
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
The total product of labor
Total Output
500
20,000-square-f oot plant
400
10,000-square-foot plant
300
200
100
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140
Number of Workers
The marginal product of labor when plant size is
10,000 square feet
Marginal Product
5.0
4.0
3.0
2.0
1.0
0
10
20
30
40
50
60
70
80
90 100 110 120 130 140
–1.0
–2.0
Number of Workers
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 thumb
The 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
Workers
10
20
30
40
50
60
70
80
90
100
110
120
130
140
Total
Marginal
Marginal
Marginal
Product Product Revenue Product
Cost
93
160
135
4.2
168
160
180
4.5
180
160
230
5
200
160
263
3.3
132
160
293
3
120
160
321
2.8
112
160
346
2.5
100
160
368
2.2
88
160
388
2.0
80
160
400
1.2
48
160
403
0.3
12
160
391
-1.2
-48
160
380
-1.1
-44
160
Problem
Let 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 returns
1. 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 dimensions
Example: 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?