Transcript Finance 30210: Managerial Economics

Economics of the Firm
Cost Analysis
Primary Managerial Objective:
Minimize costs for a given production level (potentially
subject to on 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
Note: Economic Costs vs. Accounting Costs
Example: You decide to open a lemonade stand. You spend
$100 building the stand and $50 on supplies. After your first day
(you worked 8 hours), you have collected $200 in revenues
Profit (Accounting): $200 - $50 - $100
Economic profit accounts for the opportunity cost of your time and
money. Suppose that you have a bank account that earns 5%
interest annually and that you could have worked at the local
grocery for $8/hr
Profit (Economic): $200 - $50 - $100 - $150(5%/365) - $8*8
Lost
Interest
Value of
your time
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  F K , L
MPL 
Q
L
For a given production function, the marginal product of labor measures how
production responds to small changes in labor effort
Q
Q
F ( K , L)
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  .0029L
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


Quantity
Q  K .3L2  .0029L3
96.8
Labor
Maximum Production
reached at L =70
Now, let’s calculate some of the descriptive statistics

Q  K .3L2  .0029L3

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)
Recall our managerial objective:
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  .0029L3  450
Objective
Constraint


Q  K .3L2  .0029L3  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  .0029L3  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
Q
MPL 
L
MC 
w
MPL
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
5
7.1375
2.5231
1.4275 10
3.96
7.00
APL 
Q
L
AVC
AVC 
We also know that the average variable cost is related to the inverse of
average product
wL
w

Q
APL
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  .0029L3  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 
MR 
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  .0029L3
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  .0029L3
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
Now, let’s move from the short term to the long term
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. You have a production target of 450 units:
Total Costs


Minimize30K  10L
Q  K .3L2  .0029L3  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  .0029L3  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*10 + 13*10 = $430
AC = $430/450 = $0.95
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
An isoquant refers to the various combinations of inputs that generate the same
level of production
Labor
Our managerial objective is to find the
combination of inputs on this isoquant that is
associated with the lowest costs
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
Q
K
Marginal Product of Capital
Q
TRS 
Labor
L
L
MPL MPK


K Q
MPL
MPK
If you are using a lot of capital
and very little labor, TRS is big
L
Q  450
Capital
K
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
Back to the problem at hand:
Total Costs


Minimize30K  10L
Q  K .3L2  .0029L3  450
Objective
Constraint
We know one production choice that satisfies
the constraint
L = 33
K=2
TC = 30*2 + 33*10 = $390
Minimize30K  10L


Q  K .3L2  .0029L3  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

Now, let’s increase production by one unit to get
back to our initial production level by increasing
capital
Labor

Q  K .3L2  .0029L3  450
$30
Pk
MC 
MPk
212
MC = $.50
33
By altering the production process slightly, we
were able to maintain 450 units of production and
save .$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
15
Minimize30K  10L


Q  K .3L2  .0029L3  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
Minimize30K  10L


Q  K .3L2  .0029L3  450
Labor
Pk
w

MPk MPL
Pk MPK

 TRS
w MPL
Slope = TRS
22
Q  450
Capital
4
Slope = P/w
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  .0029L3  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  10L

Solution: L = 60 (K Fixed at 1)
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
Total Costs = 30 + 10(60) = $630
Average Costs = $630/450 = $1.40
Average Variable Costs = $600/450 = $1.33
MC 

Q  K .3L2  .0029L3  450
w
10

 $2.00
MPL 5
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 30(50) = $1,530
Average Costs = $1,830/450 = $3.40
MC 
w
50

 $10.00
MPL
5
Suppose that the price of labor rises to $60
Minimize30K  10L


Q  K .3L2  .0029L3  450
Pk
30

 .76
MPk 39
Labor
w
50

 .80
MPL 62
In the long run, if
your production
technique is
flexible, you can
avoid cost
increases!
22
13
Q  450
Capital
4
10
The elasticity of substitution measures curvature of the production function
(flexibility of production)
L
%  
K


% TRS
'
L
L
  8
K
TRS=12
L
 4
K
TRS=9
K
 L  84
%   
*100  100%
4
K
12  9
%TRS 
*100  33%
9
100

3
33
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  .0029L3

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 ( 2 K ,2 L)  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 ( 2 K ,2 L )  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 ( 2 K ,2 L)  2 F ( K , L)
Costs
MC = AC
y
Marginal costs are always equal to average costs
Costs are constant (size doesn’t matter)
Estimating Production Functions


Q  F ( K , L)  AK L
%Q   %K    %L
Labor Growth
Capital Growth
Output Growth
Multifactor
Productivity
Growth
%A  %Q   %K    %L
Example: Estimating Production Functions


y  AK L
A Cobb-Douglas Production function was estimated for the
aggregate production sector of the US
y  AK L
.30 .63
Average Annual Growth = 1.5%
   1
Example: Estimating Productions

 
y  AK Lp Lnp
Non-Production Labor
Production Labor



Food/Beverage
.555
.439
.076
1.070
Textiles
.121
.549
.335
1.004
Furniture
.205
.802
.103
1.109
Petroleum
.308
.546
.089
.947
Stone, Clay, etc.
.632
.032
.366
1.029
Primary Metals
.371
.077
.509
.958
Industry
   