Production Function and Cost Analysis

Production Function
Q = aLbK1-b or c
– b+c > 1 IRTS
– b+c = 1 CRTS
– b+c < 1 DRTS
– Short Run Analysis: MPK = c Q/K &
MPL = b Q/L
– b & c are elasticities of K & L factors
– LogQ=loga+blogL+clogK + dlogT
where T  technology
2003 Prentice Hall Business Publishing Managerial Economics, 4/e Keat/Young

The Theory and
Estimation of Cost
• Definition of Cost
• The Short Run Relationship Between
Production and Cost
– The Short Run Cost Function
• The Long Run Relationship Between
Production and Cost
– The Long Run Cost Function
• The Learning Curve

Definition of Cost
• A cost is relevant if it is affected by a
management decision.
– Historical cost is incurred at the time of
procurement
– Replacement cost is necessary to replace
inventory
• Are historical costs relevant?

Definition of Cost
• There are two types of cost associated with
economic analysis
– Opportunity cost is the value that is forgone in
choosing one activity over the next best
alternative
– Out-of-pocket cost is actual transfer of value that
occur
• Which cost is relevant?

Definition of Cost
• There are two types of cost associated with
time
– Incremental cost varies with the range of options
available in the decision making process.
– Sunk cost does not vary with decision options.
• Is sunk cost relevant?

SR Relationship Between
Production and Cost
• A firm’s cost structure is related to its
production process.
– Costs are determined by the production
technology and input prices.
• Assuming that the firm is a “price taker” in the input
market.

Production and Cost
• Total variable cost
(TVC) is associated
with the variable
input
– Assume w=$500 per
unit (price-taker)
Total
Input
(L) Q (TP) MP
TVC
(wL)
0 0 0
1 1,000 1,000 500
2 3,000 2,000 1,000
3 6,000 3,000 1,500
4 8,000 2,000 2,000
5 9,000 1,000 2,500
6 9,500 500 3,000
7 9,850 350 3,500
8 10,000 150 4,000
9 9,850 -150 4,500

Production and Cost
• TP and TVC are mirror images of each
other
Kings Dominion
Example

Production and Cost
• Total cost (TC) is the cost associated with all
of the inputs. It is the sum of TVC and TFC.
– TC=TFC+TVC
• Marginal Costs
• Average Costs Tool Set for
Production Cost
Analysis
vs.
Production Process
Analysis

Production and Cost
• Marginal cost (MC) is the change in total
cost associated a change in output.
MC TC
Q
= D
D
TVC
TFC
TFC TVC
MC TC
= D ( ) 0
= + D
+ D
= D
= D +
TVC
Q
Q
Q
Q
Q
D
D
D
D
D

Production and Cost
• Add marginal
cost to the
table
Total
Input
(L) Q MP
TVC
(wL) MC
0 0 0
1 1,000 1,000 500 0.50
2 3,000 2,000 1,000 0.25
3 6,000 3,000 1,500 0.17
4 8,000 2,000 2,000 0.25
5 9,000 1,000 2,500 0.50
6 9,500 500 3,000 1.00
7 9,850 350 3,500 1.43
8 10,000 150 4,000 3.33
9 9,850 -150 4,500

Production and Cost
• Observe that:
– When MP is
increasing,
MC is
decreasing.
– When MP is
decreasing,
MC is
increasing.
Total
Input
(L) Q MP
TVC
(wL) MC
0 0 0
1 1,000 1,000 500 0.50
2 3,000 2,000 1,000 0.25
3 6,000 3,000 1,500 0.17
4 8,000 2,000 2,000 0.25
5 9,000 1,000 2,500 0.50
6 9,500 500 3,000 1.00
7 9,850 350 3,500 1.43
8 10,000 150 4,000 3.33
9 9,850 -150 4,500

Production and Cost
• The relationship between MP and MC is
MP
MC = D TVC = w ·D
L
= w · D
L
= · 1
=
w
MP
w
Q
Q
Q
D
D
D
Law of diminishing returns implies that
MC will eventually increase! Why?

The Short Run Cost Function
• Average total cost (ATC) is the average per-unit
cost of using all of the firm’s inputs (TC/Q)
– Average variable cost (AVC) is the average per-unit
cost of using the firm’s variable inputs
(TVC/Q)
– Average fixed cost (AFC) is the average per-unit
cost of using the firm’s fixed inputs (TFC/Q)

• Add ATC = AFC + AVC to the table

• ATC = AFC + AVC

• Production cost graph or map is

• Important Map Observations
– AFC declines steadily over the range of
production. Why?
– In general, ATC is u-shaped. Why?
– MC intersects the minimum point (q*) on ATC.
Why?

• Important Map Observations
– What is the economic significance of q*?

• Average total cost (ATC) is the average per-unit
cost of using all of the firm’s inputs (TC/Q)
–At Q* - ATC is minimized or inputs are
used most efficiently given the
production function
Going at 55 MPH

• A change in input
prices will shift the
cost curves.
– If fixed input costs are
reduced then ATC will
shift downward. AVC
and MC will remain
unaffected.
Computer Chip
Case

• A change in input
prices will shift the
cost curves.
– If variable input
costs are reduced
then MC, AVC, and
AC will all shift
downward.
Airline Industry
Case

• Yahoo Group
Discussion
– What is different
about dot.com
businesses?
Irrational
Exuberance

The LR Relationship Between
Production and Cost
• In the long run, all inputs are variable.
– What makes up LRAC?

The Long-Run Cost Function
• LRAC is made up for
SRACs
– SRAC curves
represent various
plant sizes
– Once a plant size is
chosen, per-unit
production costs are
found by moving
along that particular
SRAC curve

• The LRAC is the lower envelope of all of the
SRAC curves.
– Minimum efficient scale is the lowest output level
for which LRAC is minimized
Is LRAC a function of market size?
What are implications?

• Reasons for Economies of Scale…
Increasing returns to scale
Specialization in the use of labor and capital
• Economies in maintaining inventory
• Discounts from bulk purchases
• Lower cost of raising capital funds
• Spreading promotional and R&D costs
Management efficiencies

• Reasons for Diseconomies of Scale…
 Decreasing returns to scale
 Input market imperfections
 Management coordination and control
problems

The Learning Curve
• Measures the
percentage decrease in
additional labor cost
each time output
doubles.
– An “80 percent” learning
curve implies that the
labor costs associated
with the incremental
output will decrease to
80% of their previous
level.

The Learning Curve
• A downward slope in the learning curve
indicates the presence of the learning curve
effect
– Why? Workers improve their productivity with
practice
• The learning curve effect shifts the SRAC
downward

Production Function and Cost Analysis

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