Finance 30210: Managerial
Economics
Cost Analysis
Here’s the overall objective for the supply side
Production
Decisions
Product
Markets
Factor
Markets
Supply/Demand Determines
Factor prices (We have
covered this)
Factor Usage/Prices
Determine Production
Costs (We are here
now)
Demand determines
markup over costs
(Coming soon!)
Primary Managerial Objective:
Minimize costs for a given production level (potentially
subject to one or more constraints)
Example: PG&E would like to meet the daily
electricity demands of its 5.1 Million
customers for the lowest possible cost
Or
Maximize production levels while operating within a
given budget
Example: George Steinbrenner and would
like to maximize the production of the NY
Yankees while staying within the salary cap
The starting point for this analysis is to think carefully about where
your output comes from. That is, how would you describe your
production process
“is a function of”
Q  F  X 1 , X 2 , X 3 ,...
Production Level
One or more inputs
A production function is an attempt to describe what inputs are
involved in your production process and how varying inputs affects
production levels
Note: We are not trying to perfectly match
reality…we are only trying to approximate it!!!
Some production processes might be able to be described fairly easily:
Booth(s)
Sugar
(Lbs)
Your Time
(Minutes)
Q  F B, L, S ,W , C, T 
8 Oz. Glasses
of Lemonade
Lemons
Water
(Gallons)
Paper
Cups
With a fixed recipe for lemonade, this will probably be a very linear
production process
Lemonade recipe (per 8oz glass)
•Squeeze 1 Lemon into an 8 oz glass
•Add 2 oz. of Sugar
•Add 8 oz. of Water
•Stir for 1 minute to mix
We need a
booth to sell
the
lemonade!
2 oz for each
glass times
16 glasses =
2 lbs
16 Cups will
be needed
16  F 1,16,2,1,16,16
16 glasses
available for
sale
1 Lemon per
glass
1 Gallon of
Water
Makes 16 8
oz glasses
1 minute per
glass to stir
each 8 oz
glass
In fact, we could write the production function very
compactly:
Lemonade recipe (per 8oz glass)
•Squeeze 1 Lemon into an 8 oz glass
•Add 2 oz. of Sugar
•Add 8 oz. of Water
•Stir for 1 minute to mix
Q  I ( B) * X
Indicator Function
0, if B  0
I ( B)  
 1, if B  1
# of Lemonade “Kits”
(one “kit” = 1 Lemon, 2oz.
Sugar, 8 oz. Water, 1
Minute)
Or, we can look at this graphically
Q  I ( B) * X
Q
B 1
Glasses
available for
sale
Slope = 1
B0
X
# of Lemonade “Kits”
(one “kit” = 1 Lemon, 2oz.
Sugar, 8 oz. Water, 1
Minute)
Some production processes might be more difficult to specify:
How would you
describe the production
function for the
business school?
Q  F  X1 , X 2 ,...
Input(s)
Output(s)
How would you
describe the production
function for the
business school?
What is the “product” of Mendoza College of Business? YOU ARE!
Finance
Degrees
Undergraduate (BA)
Accounting
1 Year MBA (MBA)
Marketing
2 Year MBA (MBA)
Management
South Bend EMBA (MBA)
Chicago EMBA (MBA)
Masters of Accountancy (MA)
Masters of Nonprofit Administration (MA)
How would you describe
the production function for
the business school?
How would you characterize the “inputs” into Mendoza College of Business
Facilities
•Classroom Space
•Office Space
•Conference/Meeting Rooms
Equipment
•Information Technologies
•Communications
•Instructional Equipment
Capital
Inputs
Personnel
•Faculty (By Discipline)
•Administrative
•Administrative Support
•Maintenance
Labor Inputs
Staff
How would you
describe the production
function for the
business school?
Have we left out an output?
Notre Dame, like any other university, is involved in both the
production of knowledge (research) as well as the distribution of
knowledge (degree programs)
 Degrees 
Research   F Capital , Labor 


Should the two outputs be treated as
separate production processes?
The next question would be: What is your ultimate objective?
 Degrees 
Research   F Capital , Labor 


Is Notre Dame trying to maximize the quantity and quality of
research and teaching while operating within a budget?
OR
Is Notre Dame trying to minimize costs while maintaining
enrollments, maintaining high research standards and a top
quality education?
Does it
matter?
The Notre
Dame
Decision
Tree
School of
Architecture
Under the golden dome, resources are allocated
across colleges to maximize the value of Notre
Dame taking into account enrollment projections,
research reputation, education quality, and
endowment/resource constraints
College of Arts &
Letters
College of
Business
School of
Architecture
School of
Architecture
Given the resources handed down to her, Dean
Woo allocates resources across departments to
maximize the value of a Business Degree and to
maximize research output.
Finance
Department
Management
Department
Marketing
Department
Accounting
Department
Graduate
Programs
Department chairs receive resources from Dean Woo and allocate those resources to
maximize the output (research and teaching) of the department
Another issue has to do with planning horizon.
Different resources are treated as unchangeable (fixed) over
various time horizons
It might take 5 years to
design/build a new
classroom building
It could take 6 months
to install a new
computer network
Now
6 mo
1 yr
2 yr
5 yr
10 yr
It takes 1 year to hire a
new faculty member
Tenured faculty are essentially can’t be let go
Shorter planning horizons will involve more factors that will be considered fixed
From here on, lets keep things as simple as possible…
You produce a single output. There is no distinction as far as quality is concerned, so
all we are concerned with is quantity. You require two types of input in your production
process (capital and labor). Labor inputs can be adjusted instantaneously, but capital
adjustments require at least 1 year
Total
Production
“Is a
function of”
Q  F K , L
Capital (Fixed
for any
planning
horizon under
1 year
Labor (always
adjustable)
Q  F K , L
Some definitions
Marginal Product: marginal product measures the change in total production
associated with a small change in one factor, holding all other factors fixed
MPL 
Q
L
Q
MPK 
K
Average Product: average product measures the ratio of input to output
Q
APL 
L
Q
APK 
K
Elasticity of Production: marginal product measures the change in total
production associated with a small change in one factor, holding all
other factors fixed
%Q MPL
L 

%L APL
K 
%Q MPK

%K APK
Over a short planning horizon, when many factors are considered fixed (in
this case, capital), the key property of production is the marginal product of
labor.
Q
MP

L
Q  F K, L
L


For a given production function, the marginal product of labor measures how
production responds to small changes in labor effort
Q
Fll ( K , L)  0
Q
F ( K , L)
Fll ( K , L)  0
F ( K , L)
OR
L
Diminishing Marginal Returns: As labor
input increases, production increases,
but at a decreasing rate
L
Increasing Marginal Returns: As labor
input increases, production increases,
but at an increasing rate
Consider the following numerical example:

Q  K .3L  .0029 L
2
3

We start with a production function
defining the relationship between
capital, labor, and production
Capital is fixed in the short run.
Let’s assume that K = 1

Q  1 .3L2  .0029 L3

Suppose that L = 20.
 
 
Q  1 .3 202  .0029 203  96.8

Q  K .3L2  .0029 L3

Quantity
Increasing
Marginal Returns
Decreasing
Marginal Returns
Negative
Marginal Returns
96.8
Labor
Maximum Production
reached at L =70
Now, let’s calculate some of the descriptive statistics

Q  K .3L2  .0029 L3

MPL 
Q
L
Q
APL 
L
Recall, K = 1
Labor (L)
Quantity (Q)
MPL
APL
Elasticity
0
0
---
---
---
1
.2971
.2971
.2971
1
2
1.1768
.8797
.5884
1.495
3
2.6217
1.4449
.8739
1.653
4
4.6114
1.9927
1.1536
1.727
5
7.1375
2.5231
1.4275
1.7674
MPL
L 
APL
The properties of the marginal product of labor will determine the
properties of the other descriptive statistics
MP hits a maximum
at L = 35
1
Elasticity of
production greater
than one indicates
MP>AP (Average
product is rising)
MPL
L 
1
APL
Elasticity of
production less than
one indicates
MP<AP (Average
product is falling)
Cost Minimization: Short Run
The cost function for the firm can be written as
Total Costs  rk  wl
Capital costs are
fixed in the short
run!
Given the costs of the firm’s inputs, the problem facing the firm is to
find the lowest cost method of producing a fixed amount of output
Min rk  wl
l
subject to
F (k , l )  Q
k
Cost Minimization: Short Run
is fixed
(l )  rk  wl   F k , l   Q 
Remember…this needs to be positive!!
First Order Necessary Conditions
 l (l )  w  Fl (k , l )  0
w

Fl ( k , l )
Q  F (k , l )
F (k , l )  Q
Marginal costs refer to changes in total costs when
production increases
d rk  wl 
MC 
dQ
With capital fixed, marginal
costs are only influenced by
labor decisions in the short
run
Average (Unit) costs refer to total costs divided by
total production
rk  wl  rk   wl 
AC 
     
Q
Q Q
Average fixed costs fall as output increases
k
Cost Minimization: Short Run
is fixed
(l )  rk  wl   F k , l   Q 
w

Fl ( k , l )
F (k , l )  Q
Recall that lambda measures the
marginal impact of the constraint.
In this case, lambda represents the
marginal cost of producing more
output
Fll (k , l )  0
Fll (k , l )  0
Marginal costs
are increasing
Marginal costs
are decreasing
Fll (k , l )  0
Marginal Cost vs. Average Cost
Costs
AC
Minimum AC
occurs where
AC=MC
MC
y
When AC is greater
than MC, AC Falls
When AC is less than
MC, AC rises
Fll (k , l )  0
Marginal Cost vs. Average Cost
Costs
AC
MC
y
If production exhibits increasing marginal productivity, then Average
Costs decline with production (it pays to be big!)
Back to our example:
Minimize costs for a given production level (potentially
subject to on or more constraints)
Let’s imagine a simple environment where you can take the cost
of labor as a constant. Suppose that labor costs $10/hr and that
you have one unit of capital with overhead expenses of $30. You
have a production target of 450 units:
=1


Minimize30  10L
Q  K .3L2  .0029 L3  450
Objective
Constraint


Q  K .3L2  .0029 L3  450
With only one variable factor, there is
no optimization. The production
constraint determines the level of the
variable factor.
Quantity
450
Labor
450 Units of production
requires 60 hours of labor
(assuming that K=1)
Let’s imagine a simple environment where you can take the cost of labor as a
constant. Suppose that labor costs $10/hr and that you have one unit of capital
with overhead expenses of $30. You have a production target of 450 units:
=1
Total Costs


Minimize30  10L
Q  K .3L2  .0029 L3  450
Objective
Constraint
Solution: L = 60
Total Costs = 30 + 10(60) = $630
Average Costs = $630/450 = $1.40
Average Variable Costs = $600/450 = $1.33
Suppose that you increase
your production target to
451. How would your costs
be affected?
If the marginal product of labor measures output per unit labor, then the
inverse measures labor required per unit output
w
Q
MC

MPL 
MPL
L
Labor
(L)
Quantity
(Q)
MPL
APL
W
MC
0
0
---
---
---
---
1
.2971
.2971
.2971
10
33.65
33.65
2
1.1768
.8797
.5884
10
11.36
16.99
3
2.6217
1.4449
.8739
10
6.92
11.44
4
4.6114
1.9927
1.1536 10
5.01
8.66
60
450
4.68
7.5
2.13
1.33
10
Q
APL 
L
AVC
wL
w
AVC 

Q
APL
We also know that the average variable cost is related to the inverse of
average product
Properties of
production
translate directly
to properties of
cost
MC<AVC. Average
Variable Cost is
falling
MC>AVC. Average
Variable Cost is
Rising
MC hits a minimum
at L = 35
Labor
Elasticity of
production greater
than one indicates
MP>AP (Average
product is rising)
MPL
L 
1
APL
Elasticity of
production less than
one indicates
MP<AP (Average
product is falling)
For now, we are only dealing with the cost side, but eventually, we will be
maximizing profits.
=1
Total Costs


Minimize30  10L
Q  K .3L2  .0029 L3  450
Objective
Constraint
We just minimized costs of one particular production target. Maximizing profits
involves varying the production target (knowing that you will minimize the costs of
any particular target). There should be one unique production target that is
associated with maximum profits:
Maximum Profits
MR 
MR  MC
TR
Q
MC 
MC 
TC
Q
w
MPL
MR * MPL  w
Optimal
Factor
Use
Recall the alternative management objective:
Maximize production levels while operating within a
given budget
Let’s imagine a simple environment where you can take the cost of labor
as a constant. Suppose that labor costs $10/hr and that you have one
unit of capital with overhead expenses of $30. You have a production
budget of $630:
Total Output
Available budget

Maximize K .3L2  .0029 L3
Objective

30  10L  630
Constraint
30  10L  630
Just like before, there is no
optimization. The budget
constraint determines the level of
the variable factor.
Cost
630
Labor
$630 budget restricts you to 60
hours of labor (assuming that
overhead = $30)
Total Output
Available budget

Maximize K .3L2  .0029 L3
Objective

30  10L  630
Constraint
Now, if we were to think about altering the objective we would be
considering the effect on production of a $1 increase in the budget:
Change in
production
Now, take the profit maximizing
condition and flip it
Q
1
MPL


TC MC
w
1
1

MR MC
Change in
Budget
Both managerial objectives yield
the identical result!!!
MR * MPL  w
Optimal
Factor Use
An isoquant refers to the various combinations of inputs that generate the
same level of production
Labor
L = 33
L = 13
Q  450
K=2
K = 30
Capital
A key property of production in the long run has to do with the
substitutability between multiple inputs.
Labor
L
TRS 
K
L
Q  450
Capital
K
The Technical rate of substitution (TRS) measures the amount of one input required
to replace each unit of an alternative input and maintain constant production
Recall some earlier definitions:
MPL 
Q
L
MPK 
Marginal Product of Labor
TRS 
Labor
Q
K
Marginal Product of Capital
MPK
MPL
L
If you are using a lot of
capital and very little labor,
TRS is small
L
Q  450
Capital
K
K
k
Long Run Characteristics
l
l
 
k
is variable
'
l
%  
k


%TRS
l
 
k
The elasticity of substitution
measures curvature of the
production function (flexibility of
production)
k
Technical rate of Substitution measures the degree in which you can alter
the mix of inputs in production. Consider a couple extreme cases:
Perfect substitutes can
always be can always be
traded off in a constant ratio
Labor
Perfect compliments have no
substitutability and must me
used in fixed ratios
Labor
Q  20
Q  20
Capital
Capital
Cost Minimization: Long Run
Minrk  wl
k ,l
subject to
k
is variable
F (k , l )  Q
(l , k )  rk  wl   F k , l   Q 
Cost Minimization: Long Run
(l , k )  rk  wl   F k , l   Q 
First Order Necessary Conditions
 l (l ,  )  w  Fl (k , l )  0
 k (l ,  )  r  Fk (k , l )  0
r Fk (k , l )

w Fl (k , l )
w
r


Fl (k , l ) Fk (k , l )
Q  F (k , l )
Again, back to our example
Let’s imagine a simple environment where you can take the cost of
labor and the cost of capital as a constant. Suppose that labor costs
$10/hr and that capital costs $30 per unit. You have a production
target of 450 units:
Total Costs


Minimize 30K  10L
Q  K .3L2  .0029 L3  450
Objective
Constraint
Now we have two variables to solve for instead of just
one!
Consider two potential choices for Capital and Labor


Q  K .3L2  .0029 L3  450
L = 33
K=2
TC = 30*2 + 33*10 = $390
AC = $390/450 = $0.86
This procedure
is relatively
labor intensive
L = 13
K = 30
TC = 30*30 + 13*10 = $1030
AC = $1030/450 = $2.29
This procedure
is relatively
capital intensive
With more than one input, there should be multiple combinations of
inputs that will produce the same level of output
Minimize 30K  10L


Q  K .3L2  .0029 L3  450
Suppose that we lowered production by 1 unit by
decreasing labor. What would happen to costs?
Labor
$10
Total Cost = 30*2 + 33*10 = $390
MC 
33
w
MPL
20
MC = $.50
Q  450
Capital
2
Minimize 30K  10L


Q  K .3L2  .0029 L3  450
Now, let’s increase production by one unit to get back to our
initial production level by increasing capital
$30
Pk
MC 
MPk
Labor
212
MC = $.50
33
By altering the production process slightly, we
were able to maintain 450 units of production
and save $0.36!
MC = $.14
Q  450
Capital
2
Here, we have too
much labor. We
can save costs by
substituting capital
for labor
Pk
30

 .14
MPk 212
w
10

 .50
MPL 20
Here, we have too much capital.
We can save costs by substituting
labor for capital
Labor
Pk
30

 1.11
MPk 27
33
w
10

 .12
MPL 86
11
Q  450
Capital
2
1
5
Minimize 30K  10L


Q  K .3L2  .0029 L3  450
Pk
30

 .28
MPk 106
Labor
w
10

 .27
MPL 36
Total Cost = 30*4 + 10*22 = $340
Average Cost = $.75
22
Q  450
Capital
4
Short Run vs. Long Run
Minimize 30K  10L
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 10(60) = $630
Average Costs = $630/450 = $1.40
Average Variable Costs = $600/450 = $1.33
w
MC 
MPL


Q  K .3L2  .0029 L3  450
Solution: L = 22, K = 4
Total Cost = 30*4 + 10*22 = $340
Average Cost = $.75
MC 
Pk
w

MPK MPL
Long Run Average Cost will always be less than or
equal short run average costs due to the increased
flexibility of inputs
Each point on the long run average cost curve should represent the
minimum of some short run average cost curve
Average Cost
SRAC
SRAC
SRAC
SRAC
LRAC
$1.40
$0.75
Quantity
450
Suppose that the price of labor rises to $50
Minimize30K  50L
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 10(60) = $630
Average Costs = $630/450 = $1.40
Average Variable Costs = $600/450 = $1.33
w
10
MC 

 $2.00
MPL 5


Q  K .3L2  .0029 L3  450
In the short run, factor
price changes can’t be
avoided without affecting
the production target, so
costs are very sensitive to
factor price changes
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 50(60) = $3,030
Average Costs = $3,030/450 = $6.73
w
50
MC 

 $10.00
MPL
5
Suppose that the price of labor rises to $50
Minimize30K  50L

Pk
30

 .76
MPk 39
Labor

Q  K .3L2  .0029 L3  450
w
50

 .80
MPL 62
In the long run, if
your production
technique is
flexible, you can
avoid cost
increases!
Total Costs = 30(10) + 50(13) = $950
Average Costs = $630/450 = $2.11
22
13
Q  450
Capital
4
10
Elasticity of substitution determines the response of costs to
changes in input prices
mc
w
l
Low elasticity of
substitution
means that
production is very
inflexible
w
l
k
Low price elasticity
means that factor
demands don’t
respond to factor
prices
Costs are very
sensitive to factor
price changes
Elasticity of substitution determines the response of costs to
changes in input prices
mc
w
l
k
High elasticity of
substitution
means that
production is very
flexible
l
High price elasticity
means that factor
demands respond
significantly to
factor prices
w
Costs are very
insensitive to factor
price changes
As you expand production in the long
run, you are adjusting both factors, so
your costs will not depend on marginal
products!
Labor
Q  600
22
Q  550
Q  500
Q  450
Capital
4
In the long run, we are not looking for increasing or decreasing
marginal returns, but instead, we are looking for increasing or
decreasing returns to scale
Recall the production function we have been working with.

Q  K .3L2  .0029 L3

1 Unit of capital and 20 units of labor generate 96.8 units of output.
 
 
Q  1 .3 202  .0029 203  96.8
Suppose we double our inputs
 
 
Q  2 .3 402  .0029 403  588
Doubling the inputs more than
doubles production! We call
this increasing returns to scale
Increasing Returns to Scale
F (2k ,2l )  2 F (k , l )
Costs
AC
MC
y
Marginal costs are always less than average costs
Costs are decreasing (it pays to be big)
F (2k ,2l )  2 F (k , l )
Decreasing returns to Scale
Costs
MC
AC
y
Marginal costs are always greater than average costs
Costs are increasing (it pays to be small)
Constant Returns to Scale
F (2k ,2l )  2 F (k , l )
Costs
MC = AC
y
Marginal costs are always equal to average costs
Costs are constant (size doesn’t matter)