Chapter 8
Cost
McGraw-Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.
Main Topics
Types of cost
What do economic costs include?
Short-run cost: one variable input
Long-run cost: cost minimization with two
variable inputs
Average and marginal costs
Effects of input price changes
Economies and diseconomies of scale
8-2
Types of Cost
Firm’s total cost is the expenditure required to
produce a given level of output in the most
economical way
Variable costs are the costs of inputs that vary
with output level
Fixed costs do not vary as the level of output
changes, although might not be incurred if
production level is zero
Avoidable versus sunk costs
8-3
Production Costs: An Example
Table 8.1: Fixed, Variable, and Total Costs of Producing
Garden Benches
Number of
Benches
Produced per
Week
Fixed Costs
(per Week)
Variable Cost
(per Week)
Total Cost
(per Week)
0
$1,000
$0
$1,000
33
$1,000
500
1,500
74
$1,000
1,000
2,000
132
$1,000
2,000
3,000
8-4
Economic Costs
Some economic costs are hidden, such
as lost opportunities to use inputs in
other ways
Example: Using time to run your own firm
means giving up the chance to earn a salary
in another job
An opportunity cost is the cost
associated with forgoing the opportunity
to employ a resource in its best
alternative use
8-5
Short Run Cost:
One Variable Input
If a firm uses two inputs in production, one is
fixed in the short run
To determine the short-run cost function with
only one variable input:
Identify the efficient method for producing a given
level of output
This shows how much of the variable input to use
Firm’s variable cost = cost of that amount of input
Firm’s total cost = variable cost + any fixed costs
Can be represented graphically or
mathematically
8-6
Figure 8.1: Variable Cost from
Production Function
8-7
Figure 8.2: Fixed, Variable, and
Total Cost Curves
Dark red curve is
variable cost
Green curve is fixed
cost
Light red curve is
total cost, vertical
sum of VC and FC
8-8
Long-Run Cost: Cost Minimization
with Two Variable Inputs
In the long run, all inputs are variable
Firm will have many efficient ways to
produce a given amount of output, using
different input combinations
Which efficient combination is cheapest?
Consider a firm with two variable inputs K
and L, and inputs and outputs that are
finely divisible
8-9
Isocost Lines
 An isocost line connects all input combinations with the
same cost
 If W is the cost of a unit of labor and R is the cost of a
unit of capital, the isocost line for total cost C is:
WL  RK  C
 Rearranged,
 C  W
K   
R  R

L

 Thus the slope of an isocost line is –(W/R), the
negative of the ratio of input prices
8-10
Isocost Lines, continued
Isocost lines closer to the origin represent
lower total cost
A family of isocost lines contains, for given
input prices, the isocost lines for all possible
cost levels of the firm
Note the close relationship between isocost
lines and consumer budget lines
Lines show bundles that have same cost
Slope is negative of the price ratio
8-11
Sample Problem 1:
Plot the isocost line for a total cost of
$20,000 when the wage rate is $10 and the
rental rate is $40.
How does the isocost line change if the
wage rises to $20?
Least-Cost Production
How do we find the least-cost input
combination for a given level of output?
Find the lowest isocost line that touches the
isoquant for producing that level of output
No-Overlap Rule: The area below the isocost
line that runs through the firm’s least-cost input
combination does not overlap with the area
above the Q-unit isoquant
Again, note the similarities to the consumer’s
problem
8-13
Garden Bench Example,
Continued
In the long run, Naomi and Noah can
vary the amount of garage space they
rent and the number of workers they hire
An assembly worker earns $500 per
week
Garage space rents for $1 per square
foot per week
Inputs are finely divisible
8-14
Figure 8.7: Least-Cost Method,
No-Overlap Rule Example
Square Feet
of Space, K
A
2500
2000
D
1500
B
1000
Q = 140
C = $3500
500
C = $3000
1
2
3
4
5
6
Number of Assembly
Workers, L
8-15
Interior Solutions
A least-cost input combination that uses at
least a little bit of every input is an interior
solution
Interior solutions always satisfy the tangency
condition: the isocost line is tangent to the
isoquant there
Otherwise, the isocost line would cross the isoquant
Create an area of overlap between the area under
the isocost line and the area above the isoquant
This would not minimize the cost of production
8-16
Least-Cost Production and MRTS
 Restate the tangency condition in terms of marginal
products and input prices:
 Slope of isoquant = -(MRTSLK)
 MRTS = ratio of marginal products
 Slope of isocost lines = -(W/R)
 Thus the tangency condition says:
MPL W

MPK
R
or
MPL MPK

W
R
 Marginal product per dollar spent must be equal
across inputs when the firm is using a least-cost input
combination
8-17
Least-Cost Input Combination
 How can we find a firm’s least-cost input combination?
 If isoquant for desired level of output has declining
MRTS:
 Find an interior solution for which the tangency condition
formula holds
 That input combination satisfies the no-overlap rule and must
be the least-cost combination
 If isoquant does not have declining MRTS:
 First identify interior combinations that satisfy the tangency
condition, if any
 Compare the costs of these combinations to the costs of any
boundary solutions
8-18
Sample Problem 2:
Suppose the production function for
Gadget World is Q = 5L0.5K0.5. The wage
rate is $25 and the rental rate is $50.
What is the least-cost combination of
producing 100 gadgets? 200?
The Firm’s Cost Function
To determine the firm’s cost function need to
find least-cost input combination for every
output level
Firm’s output expansion path shows the
least-cost input combinations at all levels of
output for fixed input prices
Firm’s total cost curve shows how total cost
changes with output level, given fixed input
prices
8-20
Figure 8.10: Output Expansion
Path and Total Cost Curve
8-21
Average and Marginal Cost
 A firm’s average cost, AC=C/Q, is its cost per unit of
output produced
 Marginal cost measures now much extra cost the
firm incurs to produce the marginal units of output, per
unit of output added
C C Q   C Q  Q 
MC 

Q
Q
 As output increases:
 Marginal cost first falls and then rises
 Average cost follows the same pattern
8-22
Cost, Average Cost, and
Marginal Cost
Table 8.3: Cost, Average Cost, and Marginal Cost for a
Hypothetical Firm
Output (Q)
Tons per day
Total Cost (C)
(per day)
Marginal Cost
(per day)
Average Cost
(per day)
0
$0
$0
$0
1
1,000
1,000
1,000
2
1,800
800
900
3
2,100
300
700
4
2,500
400
625
5
3,000
500
600
6
3,600
600
600
7
4,300
700
614
8
5,600
1,300
700
8-23
AC and MC Curves
When output is finely divisible, can represent
AC and MC as curves
Average cost:
Pick any point on the total cost curve and draw a
straight line connecting it to the origin
Slope of that line equals average cost
Efficient scale of production is the output level at
which AC is lowest
Marginal cost:
Firm’s marginal cost of producing Q units of output
is equal to the slope of its cost function at output
level Q
8-24
Figure 8.16: Relationship
Between AC and MC
AC slopes downward
where it lies above
the MC curve
AC slopes upward
where it lies below
the MC curve
Where AC and MC
cross, AC is neither
rising nor falling
8-25
Marginal Cost, Marginal Products,
and Input Prices
Intuitively, a firm’s costs should be lower the
more productive it is and the lower the input
prices it faces
Formalize relationship between marginal cost,
marginal products, and input prices using the
tangency condition:
R
W
MC 

MPK MPL
8-26
More Average Costs: Definitions
Apply idea of average cost to firm’s variable
and fixed costs to find average variable cost
and average fixed cost:
VC
AVC 
Q
FC
AFC 
Q
Since total cost is the sum of variable and
fixed costs, average cost is the sum of AVC
and AFC:
C VC  FC VC FC
AC  


 AVC  AFC
Q
Q
Q
Q
8-27
Average Cost Curves
Fixed costs are constant so AFC is
always downward sloping
At each level of output the AC curve is
the vertical sum of the AVC and AFC
curves
Average cost curve lies above both AVC and
AFC at every output level
Efficient scale of production exceeds output
level where AVC is lowest
8-28
Figure 8.18: AC, AVC, and
AFC Curves
8-29
Figure 8.20: AC, AVC, and
MC Curves
8-30
Effects of Input Price Changes
Changes in input prices usually lead to
changes in a firm’s least-cost production
method
Responses to a Change in an Input Price:
When the price of an input decreases, a firm’s leastcost production method never uses less of that input
and usually employs more
For a price increase, a firm’s least-cost input
production method never uses more of that input
and usually employs less
8-31
Figure 8.21: Effect of an Input
Price Change
Point A is optimal
input mix when price
of labor is four times
more than the price
of capital
Point B is optimal
when labor and
capital are equally
costly
8-32
Short-run vs. Long-run Costs
 In the long run a firm can vary all inputs
 Will choose least-cost input combination for each output level
 In the short run a firm has at least one fixed input
 Produce some level of output at least-cost input combination
 Can vary output from that in short run but will have higher
costs than could achieve if all inputs were variable
 Long-run average variable cost curve is the lower
envelope of the short-run average cost curves
 One short-run curve for each possible level of output
8-33
Figure 8.24: Input Response over
the Long and Short Run
8-34
Figure 8.25: Long-run and Shortrun Costs
8-35
Figure 8.26: Long-run and Shortrun Average Cost Curves
8-36
Economies and Diseconomies of
Scale
What are the implications of returns to scale?
A firm experiences economies of scale when
its average cost falls as it produces more
Cost rises less, proportionately, than the increase in
output
Production technology has increasing returns to
scale
Diseconomies of scale occur when average
cost rises with production
8-37
Figure 8.28: Returns to Scale and
Economies of Scale
8-38
Sample Problem 3 (8.12):
Noah and Naomi want to produce 100 garden
benches per week in two production plants. The
cost functions at the two plants are
C1  600Q1  3Q
2
1
C2  650Q2  2Q
2
2
and
,
and the corresponding marginal costs are MC1 =
600 – 6Q1 and MC2 = 650 – 4Q2. What is the best
output assignment between the two plants?