Transcript Production - Arkansas State University

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).
This table [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?
Economies of increased dimensions
SA  2  rh  2  rh
2
r
h
V  r h
2
Effect of a 1 inch change in vessel
radius
Surface Area and Volume
PI
r (in.)
3.1416
6
3.1416
7
h (in.)
10
10
S.A. (sq. in.)
4146.902
4838.053
V (cu. in.)
1130.973
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 has
made shipping goods
more 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.
Definitions
Variable 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
Firm’s Costs in the Short Run
Annual Output
Total Cost Fixed Cost Variable Cost
(Thousands of Repairs)
($000s)
($000s)
(000s)
0
270.0
270
0.0
5
427.5
270
157.5
10
600.0
270
330.0
15
787.5
270
517.5
20
990.0
270
720.0
25
1207.5
270
937.5
30
1440.0
270
1170.0
35
1687.5
270
1417.5
40
1950.0
270
1680.0
45
2227.5
270
1957.5
50
2520.0
270
2250.0
55
2827.5
270
2557.5
60
3150.0
270
2880.0
T ota l C ost (T housands of D olla rs)
3,000
2,000
C ost function
1,000
0
5 1 0 1 5 20 25 30 35 40 45 50 55 6 0
O utput (T housands of Units)
Annual Output
Total Cost
(Thousands of Repairs)
($000s)
0
270.0
5
427.5
10
600.0
15
787.5
20
990.0
25
1207.5
30
1440.0
35
1687.5
40
1950.0
45
2227.5
50
2520.0
55
2827.5
60
3150.0
Ave. Cost Marginal Cost
($000s)
($000s)
85.5
60.0
52.5
49.5
48.3
48.0
48.2
48.8
49.5
50.4
51.4
52.5
31.5
34.5
37.5
40.5
43.5
46.5
49.5
52.5
55.5
58.5
61.5
64.5
C o s t/Unit (T ho us a nd s o f D o lla rs )
64
SM C
60
56
52
SAC
48
44
0
5 10 15 20 25 30 35 40 45 50 55 60
O utp ut (T ho us and s o f Units )
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:
SMC

PL
MP L
[7.1]
Recall that the marginal
product of labor will begin to
fall at some point due to the
law of diminishing returns.
Behavior of Average Fixed Cost
Fixed Cost Per Unit
300.0
As output increases, fixed
cost can be spread more
thinly
250.0
200.0
150.0
100.0
50.0
0.0
1
11
21
31
Output (Q)
41
51
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
L o ng - R un A ve r a g e C o s t
$5
SAC 1
SAC 2
( 9 ,0 0 0 - s q ua r e - ( 1 8 ,0 0 0 - s q ua r e SAC 3
fo o t p la nt)
fo o t p la nt) ( 2 7 , 0 0 0 - s q ua r e f o o t p la nt)
4
LAC = LM C
SM C 1
0
SM C 2
SM C3
72 108 144
216
O ut p ut ( T ho us a nd s o f Units )
The U-Shaped Long Run Average Cost Function
L o ng - R un A ve r a g e C o s t
LM C
LAC
SAC 1
SAC 2
SM C3
SM C 1
LM C
SAC 3
SM C 2
Decreasing returns
Increasing returns
Q m in
O utp ut
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.
QMES is large relative to he
“size of the market.” To produce on an
efficient scale, you
must supply 50% of
the product demanded
at a price equal to
minimum unit cost
Cost per unit
Demand
LAC
0
1000
2000
Q
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.
L o n g -R u n Ave ra g e C o s t
Q m in
O u tp u t
(a )
L o ng - R un A ve r a g e C o s t
Q m in
(b)
O utp ut
L o ng - R un A ve r a g e C o s t
Local telephone service,
electricity distribution, and
cable TV distribution are well
represented by this cost
function.
(c)
O utp ut