finance 30210: managerial economics cost analysis
Post on 19-Dec-2015
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TRANSCRIPT
Here’s the overall objective for the supply side
Factor Markets
Production Decisions
Product 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)
Or
Maximize production levels while operating within a given budget
Example: PG&E would like to meet the daily electricity demands of its 5.1 Million customers for the lowest possible cost
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
,...,, 321 XXXFQ Production Level
“is a function of”
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:
TCWSLBFQ ,,,,,
8 Oz. Glasses of Lemonade
Booth(s)
Lemons
Sugar (Lbs)
Water (Gallons)
Your Time (Minutes)
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
16,16,1,2,16,116 F
1 Gallon of Water Makes 16 8 oz glasses
1 minute per glass to stir each 8 oz glass
16 Cups will be needed
2 oz for each glass times 16 glasses = 2 lbs
1 Lemon per glass
We need a booth to sell the lemonade!
16 glasses available for sale
In fact, we could write the production function very compactly:
XBIQ *)(
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
# of Lemonade “Kits” (one “kit” = 1 Lemon, 2oz. Sugar, 8 oz. Water, 1 Minute)
Indicator Function
1 ,1
0 ,0)(
Bif
BifBI
Q
X
XBIQ *)(Or, we can look at this graphically
1B
0B
Slope = 1
# of Lemonade “Kits” (one “kit” = 1 Lemon, 2oz. Sugar, 8 oz. Water, 1 Minute)
Glasses available for sale
Some production processes might be more difficult to specify:
How would you describe the production function for the business school?
,..., 21 XXFQ
Output(s)Input(s)
How would you describe the production function for the business school?
What is the “product” of Mendoza College of Business? YOU ARE!
Degrees
Undergraduate (BA)
1 Year MBA (MBA)
2 Year MBA (MBA)
South Bend EMBA (MBA)
Chicago EMBA (MBA)
Finance
Accounting
Marketing
Management
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 Personnel
• Faculty (By Discipline)• Administrative • Administrative Support• Maintenance
StaffEquipment• Information Technologies• Communications• Instructional Equipment
Capital Inputs
Labor Inputs
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)
LaborCapitalF ,Research
Degrees
Should the two outputs be treated as separate production processes?
LaborCapitalF ,Research
Degrees
The next question would be: What is your ultimate objective?
Is Notre Dame trying to maximize the quantity and quality of research and teaching while operating within a budget?
Is Notre Dame trying to minimize costs while maintaining enrollments, maintaining high research standards and a top quality education?
OR
Does it matter?
School of Architecture
College of Arts & Letters
College of Business
School of Architecture
School of Architecture
Finance Department
Management Department
Marketing Department
Accounting Department
Graduate Programs
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
The Notre Dame Decision Tree
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.
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
Now 6 mo 1 yr 2 yr 5 yr 10 yr
It could take 6 months to install a new computer network
It takes 1 year to hire a new faculty member
It might take 5 years to design/build a new classroom building
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
LKFQ ,
Total Production
“Is a function of”
Capital (Fixed for any planning horizon under 1 year
Labor (always adjustable)
Some definitions LKFQ ,
Marginal Product: marginal product measures the change in total production associated with a small change in one factor, holding all other factors fixed
L
QMPL
K
QMPK
Average Product: average product measures the ratio of input to output
L
QAPL
K
QAPK
Elasticity of Production: marginal product measures the change in total production associated with a small change in one factor, holding all other factors fixed
L
LL AP
MP
L
Q
%
%K
KK AP
MP
K
Q
%
%
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.
LKFQ , L
QMPL
For a given production function, the marginal product of labor measures how production responds to small changes in labor effort
Q
L
),( LKF
Q
L
),( LKF
OR
Diminishing Marginal Returns: As labor input increases, production increases, but at a decreasing rate
Increasing Marginal Returns: As labor input increases, production increases, but at an increasing rate
0),( LKFll0),( LKFll
Consider the following numerical example:
32 0029.3. LLKQ 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
32 0029.3.1 LLQ
Suppose that L = 20.
8.96200029.203.1 32 Q
32 0029.3. LLKQ
96.8
Maximum Production reached at L =70
Labor
Qua
ntity
Increasing Marginal Returns
Decreasing Marginal Returns
Negative Marginal Returns
Now, let’s calculate some of the descriptive statistics
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
32 0029.3. LLKQ
Recall, K = 1
L
QMPL
L
QAPL
L
LL AP
MP
The properties of the marginal product of labor will determine the properties of the other descriptive statistics
1L
LL AP
MPElasticity of production less than one indicates MP<AP (Average product is falling)
Elasticity of production greater than one indicates MP>AP (Average product is rising)
1
MP hits a maximum at L = 35
Cost Minimization: Short Run
The cost function for the firm can be written as
wlkrCostsTotal 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
QlkF
tosubject
wlkrMinl
),(
Capital costs are fixed in the short run!
Cost Minimization: Short Run k is fixed
QlkFwlkrl ,)(
0),()( lkFwl ll
First Order Necessary Conditions
),( lkFQ
),( lkF
w
l
QlkF ),(
Remember…this needs to be positive!!
Marginal costs refer to changes in total costs when production increases
dQ
wlkrdMC
Q
wl
Q
kr
Q
wlkrAC
Average (Unit) costs refer to total costs divided by total production
Average fixed costs fall as output increases
With capital fixed, marginal costs are only influenced by labor decisions in the short run
Cost Minimization: Short Run k is fixed
QlkFwlkrl ,)(
),( lkF
w
l
QlkF ),(
Recall that lambda measures the marginal impact of the constraint. In this case, lambda represents the marginal cost of producing more output
0),( lkFll
0),( lkFll
Marginal costs are increasing
Marginal costs are decreasing
Marginal Cost vs. Average Cost
y
Costs
MC
AC
Minimum AC occurs where AC=MC
0),( lkFll
When AC is greater than MC, AC Falls
When AC is less than MC, AC rises
Marginal Cost vs. Average Cost
y
Costs
MC
AC
0),( lkFll
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:
4500029.3. 32 LLKQ LMinimize 1030
Objective Constraint
=1
4500029.3. 32 LLKQ
450 Units of production requires 60 hours of labor (assuming that K=1)
Labor
Qua
ntity
450
With only one variable factor, there is no optimization. The production constraint determines the level of the variable factor.
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:
4500029.3. 32 LLKQ LMinimize 1030
Objective Constraint
=1
Solution: L = 60
Total Costs
Total Costs = 30 + 10(60) = $630Average Costs = $630/450 = $1.40Average Variable Costs = $600/450 = $1.33
Suppose that you increase your production target to 451. How would your costs be affected?
Labor (L)
Quantity (Q)
MPL APL W MC AVC
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 10 2.13 1.33
L
QMPL
If the marginal product of labor measures output per unit labor, then the inverse measures labor required per unit output
L
QAPL
LAP
w
Q
wLAVC
We also know that the average variable cost is related to the inverse of average product
LMP
wMC
1L
LL AP
MPElasticity of production less than one indicates MP<AP (Average product is falling)
Elasticity of production greater than one indicates MP>AP (Average product is rising)
MC<AVC. Average Variable Cost is falling
MC>AVC. Average Variable Cost is Rising
MC hits a minimum at L = 35
Properties of production translate directly to properties of cost
Labor
For now, we are only dealing with the cost side, but eventually, we will be maximizing profits.
4500029.3. 32 LLKQ LMinimize 1030
Objective Constraint
=1Total Costs
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 MCMR
Q
TCMC
Q
TRMR
LMP
wMC wMPMR L *
Optimal Factor Use
6301030 L 32 0029.3. LLKMaximize
ObjectiveConstraint
Total Output
Maximize production levels while operating within a given budget
Recall the alternative management objective:
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:
Available budget
$630 budget restricts you to 60 hours of labor (assuming that overhead = $30)
Labor
Cos
t
630
Just like before, there is no optimization. The budget constraint determines the level of the variable factor.
6301030 L
6301030 L 32 0029.3. LLKMaximize
ObjectiveConstraint
Total Output Available budget
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:
w
MP
MCTC
Q L 1
Change in production
Change in Budget
Now, take the profit maximizing condition and flip it
MCMR
11
wMPMR L *
Optimal Factor Use
Both managerial objectives yield the identical result!!!
Capital
Labor
An isoquant refers to the various combinations of inputs that generate the same level of production
450Q
L = 33
K = 30
L = 13
K = 2
Capital
Labor
A key property of production in the long run has to do with the substitutability between multiple inputs.
450Q
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
L
K
K
LTRS
Capital
Labor
Recall some earlier definitions:
450QL
K
K
L
MPTRS
MP
L
QMPL
K
QMPK
Marginal Product of Labor Marginal Product of Capital
K
L
If you are using a lot of capital and very little labor, TRS is small
The elasticity of substitution measures curvature of the production function (flexibility of production)
k
l'
k
l
k
l
TRSkl
%
%
k is variableLong Run Characteristics
Capital
Labor
Capital
Labor
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
Perfect compliments have no substitutability and must me used in fixed ratios
20Q20Q
Cost Minimization: Long Run
QlkFwlrkkl ,),(
0),(),( lkFwl ll
First Order Necessary Conditions
),( lkFQ
0),(),( lkFrl kk ),(
),(
lkF
lkF
w
r
l
k
),(),( lkF
r
lkF
w
kl
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:
4500029.3. 32 LLKQ LKMinimize 1030
Objective Constraint
Total Costs
Now we have two variables to solve for instead of just one!
4500029.3. 32 LLKQ
Consider two potential choices for Capital and Labor
L = 33K = 2TC = 30*2 + 33*10 = $390AC = $390/450 = $0.86
L = 13K = 30TC = 30*30 + 13*10 = $1030AC = $1030/450 = $2.29
This procedure is relatively labor intensive
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
Capital
Labor
450Q
33
2
Total Cost = 30*2 + 33*10 = $390
4500029.3. 32 LLKQ LKMinimize 1030
Suppose that we lowered production by 1 unit by decreasing labor. What would happen to costs?
LMP
wMC
$10
20MC = $.50
Capital
Labor
450Q
33
2
4500029.3. 32 LLKQ LKMinimize 1030
Now, let’s increase production by one unit to get back to our initial production level by increasing capital
k
k
MP
PMC
$30
212
MC = $.50
MC = $.14
By altering the production process slightly, we were able to maintain 450 units of production and save $0.36!
Capital
Labor
450Q
33
2 15
11
14.212
30
k
k
MP
P
50.20
10
LMP
w
11.127
30
k
k
MP
P
12.86
10
LMP
w
Here, we have too much capital. We can save costs by substituting labor for capital
Here, we have too much labor. We can save costs by substituting capital for labor
Capital
Labor
450Q
4
22
4500029.3. 32 LLKQ LKMinimize 1030
28.106
30
k
k
MP
P
27.36
10
LMP
w
Total Cost = 30*4 + 10*22 = $340Average Cost = $.75
Short Run vs. Long Run
4500029.3. 32 LLKQ LKMinimize 1030
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 10(60) = $630Average Costs = $630/450 = $1.40Average Variable Costs = $600/450 = $1.33
Solution: L = 22, K = 4
Total Cost = 30*4 + 10*22 = $340Average Cost = $.75
Long Run Average Cost will always be less than or equal short run average costs due to the increased flexibility of inputs
LK
k
MP
w
MP
PMC
LMP
wMC
Average Cost
Quantity
SRAC
SRAC SRAC
SRAC
LRAC
Each point on the long run average cost curve should represent the minimum of some short run average cost curve
450
$1.40
$0.75
Suppose that the price of labor rises to $50
4500029.3. 32 LLKQ LKMinimize 5030
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 10(60) = $630Average Costs = $630/450 = $1.40Average Variable Costs = $600/450 = $1.33
00.2$5
10
LMP
wMC
Solution: L = 60 (K Fixed at 1)
Total Costs = 30 + 50(60) = $3,030Average Costs = $3,030/450 = $6.73
00.10$5
50
LMP
wMC
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
Capital
Labor
450Q
10
13
Suppose that the price of labor rises to $50
76.39
30
k
k
MP
P
80.62
50
LMP
w
4500029.3. 32 LLKQ LKMinimize 5030
In the long run, if your production technique is flexible, you can avoid cost increases!
4
22
Total Costs = 30(10) + 50(13) = $950Average Costs = $630/450 = $2.11
l
k l
w
w
mc
Elasticity of substitution determines the response of costs to changes in input prices
Low elasticity of substitution means that production is very inflexible
Low price elasticity means that factor demands don’t respond to factor prices
Costs are very sensitive to factor price changes
l
k l
w
w
mc
Elasticity of substitution determines the response of costs to changes in input prices
High elasticity of substitution means that production is very flexible
High price elasticity means that factor demands respond significantly to factor prices
Costs are very insensitive to factor price changes
Capital
Labor
450Q
22
4
As you expand production in the long run, you are adjusting both factors, so your costs will not depend on marginal products!
500Q
550Q
600Q
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
32 0029.3. LLKQ
Recall the production function we have been working with.
8.96200029.203.1 32 Q
1 Unit of capital and 20 units of labor generate 96.8 units of output.
Suppose we double our inputs
588400029.403.2 32 QDoubling the inputs more than doubles production! We call this increasing returns to scale
Increasing Returns to Scale
y
Costs
MC
),(2)2,2( lkFlkF
AC
Marginal costs are always less than average costs
Costs are decreasing (it pays to be big)
y
Costs MC
),(2)2,2( lkFlkF
AC
Decreasing returns to Scale
Marginal costs are always greater than average costs
Costs are increasing (it pays to be small)