Topic 5 – Cost Analysis / Cost Functions
Outline:
I)
II)
III)
IV)
Motivation
Short-Run Costs
Long-Run Costs
An Application
I)
Motivation / Introduction
- In analyzing the efficiency implications of various
market structures or polices, it is common to model
the firm as choosing a level of output to maximize
profits.
- By deriving the firm’s cost functions, we can
sidestep the firm’s input decisions and consider the
firm’s choice of output directly.
- This is because the tangency points between
isoquants and isocost lines tell us the total cost of
producing a given level of output in the least cost
manner.
- Thus, the firm’s cost functions subsume the input
selection decision.
- Can categorize costs in the following ways:
- Total, Average, Marginal
- Fixed vs. Variable
- We will look at the various combinations and their
relationships to one another in both the short run
and the long run.
II) Short-Run Costs (most of this should be review)
A) Total Costs
- Recall that in our two input world, total cost (TC)
is given by
TC = wL + rK
- Since some inputs are fixed in the short run (e.g.
K), there are two types of short-run costs.
(1) Fixed Costs, FC: Do not vary with output. These
are the costs of the fixed inputs.
Example: FC = rK0 when capital is fixed at K0 in
the short run.
(2) Variable costs, VC(Q) : Vary with output. These
are the costs of the variable inputs.
Example: VC(Q) = wL
Note: VC(Q) is written as a function of Q since
varying L will change Q.
Short-Run (Total) Cost Function – Defines the
minimum cost of producing each level of output
when variable inputs are used in the cost-minimizing
way.
Note: Embeds the optimal input choices we studied
in the last section.
To begin, recall that:
Short Run TC = wL + rK0
Example: Suppose capital costs $1000 per unit and
labor costs $400 per unit. Derive the costs associated
with the production technology shown in the table
below.
Fixed
Variable
Input
Input
(Capital) (Labor)
2
0
2
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
9
2
10
2
11
Output
0
76
248
492
748
1100
1416
1708
1952
2124
2200
2156
FC
VC(Q)
TC(Q)
K0 * 1000
L * 400
FC + VC(Q)
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$0
$400
$800
$1200
$1600
$2000
$2400
$2800
$3200
$3600
$4000
$4400
$2000
$2400
$2800
$3200
$3600
$4000
$4400
$4800
$5200
$5600
$6000
$6400
We can also illustrate these relationships graphically
(see diagram).
B) Average Costs
Average Fixed Costs, AFC – Fixed costs divided by
the number of units of output.
AFC 
FC
Q
- AFC declines with output
- Captures the fact that as output increases,
“overhead” expenses are spread over larger values
of Q.
Average Variable Costs, AVC – Variable costs
divided by the number of units of output.
AVC 
VC (Q)
Q
- Generally declines with output initially, then levels
off, then rises since
AVC 
VC (Q) wL w
w

 
Q
Q Q AP
L
L
Average Total Costs, ATC – Total costs divided by
the number of units of output.
ATC 
TC(Q)
Q
- Generally declines with output initially, then levels
off, then rises (for same reason as AVC).
Example: Derive average costs for previous example.
Output
0
76
248
492
748
1100
1416
1708
1952
2124
2200
FC
VC(Q)
K0 * 1000
L * 400
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$2000
$0
$400
$800
$1200
$1600
$2000
$2400
$2800
$3200
$3600
$4000
AFC
AVC
$26.32
$8.06
$4.07
$2.55
$1.82
$1.41
$1.17
$1.02
$0.94
$0.91
$5.26
$3.23
$2.44
$2.04
$1.82
$1.69
$1.64
$1.64
$1.69
$1.82
C) Marginal Costs
Marginal Cost, MC – The cost of producing an
additional unit of output.
MC 
TC (Q)
Q
- Generally declines with output initially, then levels
off, then rises since
TC (Q) VC  FC  VC wL wL w
w
MC 






Q
Q
Q Q Q Q MP
L
L
Applications:
1) Why do some, but not all, companies move
manufacturing plants to counties with lower
labor costs?
2) Why does the cost of a college education
continue to rise faster than the cost of most
manufactured goods?
Numerical Example: Derive marginal costs for
previous example.
Output
Q
TC
TC
0
76
248
492
748
1100
1416
1708
1952
2124
2200
76
172
244
292
316
316
292
244
172
76
$2000
$2400
$2800
$3200
$3600
$4000
$4400
$4800
$5200
$5600
$6000
400
400
400
400
400
400
400
400
400
400
MC =
TC/Q
5.26
2.33
1.64
1.37
1.27
1.27
1.37
1.64
2.33
5.26
Relationships Among Short-Run Cost Curves
- See diagram
(i)
The marginal cost curve intersects the ATC
and AVC cost curves at their minimum
points.
- For the same reason that MP intersects AP at
it’s maximum.
- GPA example.
(ii)
The spread between ATC and AVC
decreases as output increases.
- This is due to the fact that AFC declines with
output and ATC – AVC = AFC.
Mathematical Example
Consider the cubic cost function:
TC(Q) 100 aQ  bQ2  cQ3
Problem: Calculate TFC, TVC(Q), AFC, and
AVC(Q). Note: Calculation of MC(Q) requires the
use of calculus.
TFC 100
TVC (Q)  aQ  bQ2  cQ3
100
AFC 
Q
AVC(Q)  a  bQ  cQ2
Note: For those of you know calculus, MC(Q) is
simply the derivative of TC(Q).
III) Long Run Costs (most of this should be review)
- All inputs can be varied (no fixed costs).
A) Total Costs
- Long run total cost curve is derived from the
isoquant / isocost diagram via the output expansion
path.
Output Expansion Path – The set of cost-minimizing
input bundles at fixed input prices. Given by the
locus of tangencies between isoquants and isocosts at
different output levels.
- See diagram.
- Can use the output expansion path to create a table
relating output and total cost.
- This allows you to graph the long run total cost
curve.
- Output expansion path provides the link between
the cost curve, written as a function of output, and
the firm’s cost-minimizing input choices.
B) Average and Marginal Costs
- These are defined analogously to their short run
counterparts, except that there are no fixed costs, so
AVC = ATC
- Specifically,
ATC 
MC 
TC wL  rK

Q
Q
TC wL  rK

Q
Q
Returns to Scale and Long Run Cost Curves
- First note that if we double K and L, TC exactly
doubles, since
TC (2K, 2L) = r(2K) + w(2L)
=2 [rK + wL]
=2 TC (K, L)
a) Increasing Returns to Scale – Output more than
doubles and total cost exactly doubles, so TC(Q)
is a concave function (see diagram)
This implies that
AC 
TC
Q
is declining and that MC is below AC (see
diagram).
b) Constant Returns to Scale – Output exactly
doubles and total cost exactly doubles, so TC(Q)
is a linear function (see diagram).
This implies that
AC 
TC
Q
is constant and MC = AC (see diagram).
c) Decreasing Returns to Scale – Output less than
doubles and total cost exactly doubles, so TC(Q)
is a convex function (see diagram).
This implies that
AC 
TC
Q
is increasing and that MC is above AC (see
diagram).
Typical Case: Increasing returns at low levels of
output, followed by constant returns, followed by
decreasing returns.
 U-shaped average cost curve
IV) Application: Emergency Room Costs
- Many health policy-makers (e.g. those who
administer the Medicaid program) and hospital
administrators believe that health care costs can be
reduced by cutting down on unnecessary ER visits.
- These beliefs have lead to attempts to reduce “nonurgent” ER visits.
Question: How large are the cost savings from
reduced ER visits likely to be?
Answer: Much smaller than is commonly believed.
Reason:
The perceived costs of ER visits and hospitalizations
are typically based on average cost calculations.
These include fixed costs (e.g. costs of buildings,
equipment, and salaried staff) that will not go away if
fewer patients are served.
Example: How much cost savings could be achieved
by eliminating all “non-urgent” ER visits?
- Williams (see handout) estimates the average cost
of non-urgent ER visits to be $69 per visit.
- Actual cost savings depend on marginal costs (i.e.,
how much will total costs fall if we reduce output?)
- Williams estimates the marginal cost of non-urgent
ER visits to be $24 per visit.
- Thus, MC is only 39% of AC, so the true cost
savings will be only 39% of the projected cost
savings.