Differentiation
• Purpose- to determine instantaneous rate of
change
Eg: instantaneous rate of change in total cost per
unit of the good
We will learn
• Marginal Demand, Marginal Revenue, Marginal
Cost, and Marginal Profit
Marginal Cost : MC(q)
• What is Marginal cost?
The cost per unit at a given level of production
That is, MC(q) is the cost for an additional dinner, when
q dinners are being prepared
Marginal Analysis
• First Plan
• Cost of one more unit
• MC q  Cq  1  Cq
Dinner Example
Example 1.
We consider the cost function
C(q) = C0 + VC(q) =
$63,929.37 + 13,581.51ln(q) that was
developed in the Expenses and Profit section of
Demand, Revenue, Cost, and Profit. Recall that
a restaurant chain is planning to introduce a new
buffalo steak dinner. C(q) is the cost, in dollars,
of preparing q dinners per week for 1,000  q 
4,000.
Differentiation.
Marginal Analysis: page 2
Differentiation, Marginal
MC (q)  C (q  1)  C (q)
  63,929.37  13,581.51 ln(q  1)    63,929.37  13,581.51 ln(q) 
 13,581.51 ln(q  1)  ln(q) 
We can use a calculator or Excel to compute values of MC(q). For
example,
MC (2,000)  13,581.51 ln(2,000  1)  ln(2,000) 
 13,581.51 ln(2,001)  ln(2,000) 
 6.78906.
Thinking in terms of money, the marginal cost at the level of 2,000
dinners, is approximately $6.79 per dinner. Similar computations show that
MC(2,500)  $5.43 and MC(3,000)  $4.53.
Since the marginal cost per dinner depends upon the number of
dinners currently being prepared, it is helpful to look at a plot of MC(q) against
q. This is created in the sheet M Cost of the Excel file Dinners.xls.
Dinners.xls
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Differentiation.
Differentiation, MarginalMarginal Analysis: page 3
Marginal Cost Function, First Plan
MC(q) $ /dinner
$16
$12
$8
$4
$0
0
Dinners.xls
1,000
2,000
q dinners
3,000
4,000
(material continues)
Looking at the plot
on the left or checking
Column D in M Cost, we see
that the First Plan marginal
cost decreases considerably
as q increases. Hence, there
is an “economy of scale” as
more dinners are produced.
This is consistent with the
expectations of business
common sense.
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Marginal AnalysisMC(q) is best defined as the instantaneous rate of change in total cost,
per unit.
•
Final Plan
•
Average cost of fractionally more and fractionally less units
difference quotients
•
C q  h   C q  h 
MC q   lim
h 0
2h
•
C q  h   C q  h 
Typically use MC q  
with h = 0.001
2h
Marginal Analysis
• Ex. Suppose the cost for producing a particular
item is given by C q   8000  127q 0.607 where q
is quantity in whole units. Approximate
MC(500). h=0.001
C 500.001  C 499.999 
MC 500  
•
2  0.001
8000  127  500.0010.607  8000  127  499.9990.607

0.002
13521.77926  13521.76585

0.002
 $6.71 per unit

In terms of money, the marginal cost at the
production level of 500, $6.71 per unit
 

C(q) = $63,929.37 + 13,581.51ln(q)
Ex. Suppose the cost for producing a particular item
is given above. where q is quantity in whole
units. Approximate MC(1000) when h=0.1
Marginal Analysis
• Use “Final Plan” to determine answers
• All marginal functions defined similarly
Rq  h   Rq  h 


MR q  lim
h 0
•
2h
C q  h   C q  h 
MC q   lim
h 0
2h
Pq  h   Pq  h 
MP q   lim
h 0
2h
Differentiation, Marginal
Differentiation.
Marginal Analysis: page 8
Values for all of our marginal functions are computed in the sheets M Cost and M Profit of the Excel file
Dinners.xls. The graphs of MD(q), MR(q), and MP(q) are also displayed in those
sheets.-feb4
Marginal Demand Function
Demand Function
$0.000
$32
MD(q) $/dinner
0
D(q)
$24
$16
$8
1,000
2,000
3,000
4,000
-$0.005
-$0.010
-$0.015
$0
0
1000
2000
q
3000
4000
-$0.020
q
Many aspects of the demand function are reflected in properties of the
difference quotients for marginal demand, and in the marginal demand function.
D(q) is always decreasing. Hence, all difference quotients for marginal demand
are negative, and MD(q) is always negative. The more rapidly D(q) drops, the
more negative are the difference quotients, and the further negative is MD(q).
Dinners.xls
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Revenue Function
Differentiation, MarginalMarginal Analysis: page 9
Differentiation.
$50,000
Marginal Revnue Function
$40
$30,000
$20,000
$10,000
$0
0
1000
2000
q
3000
4000
Where the revenue function
R(q) is increasing, the difference
quotients for marginal revenue are
MR(q) $/dinner
R(q)
$40,000
$20
$0
0
1,000
2,000
3,000
4,000
-$20
-$40
q
positive, and MR(q) is positive. For example, MR(1,300) is approximately
$20. Thus, when 1,300 dinners are prepared and sold, the restaurant chain
takes in $20 more for each extra dinner. Likewise, where R(q) is decreasing,
MR(q) is negative. This shows that the maximum revenue will occur at the
value of q where the marginal revenue is equal to 0. Computations in the
sheet M Profit show that MR(2,309) = $0.01 and MR(2,310) = $0.01. Hence,
the maximum revenue occurs at either 2,309 or 2,310 dinners. Direct
computation shows that the maximum revenue is R(2,310) = $45,975.65.
Dinners.xls
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Differentiation, Marginal
Differentiation.
Marginal Analysis: page 10
Revenue
Revenue and Cost Function
$60,000
$50,000
$40,000
$30,000
$20,000
$10,000
$0
$6,000
$4,000
$2,000
$0
-$2,000 0
0
1000
2000
q
3000
4000
$0
2,000
3000
4000
q
Marginal analysis can tell us
a great deal about the profit function.
Refer back to these plots while
reading the next pages.
$10
1,000
2000
-$4,000
M Revenue
M Cost
$20
-$10 0
1000
-$6,000
Marginal Revenue & Marginal Cost
Functions
$30
$/dinner
Profit Function
P(q)
Dollars
Cost
3,000
4,000
-$20
-$30
q
Dinners.xls
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Derivatives
• Project (Marginal Revenue)
- Typically MRq  Rq
- In project,
-
MRq  1000  Rq
Recall:Revenue function-R(q)
• Revenue in million dollars R(q)
typically
MRq  Rq
1000000  Rq 
MR q  
1000
 1000 Rq 
• Why do this conversion?
Marginal Revenue in dollars per drive
15
Derivatives
• Project (Marginal Cost)



q
MC
q

C
- Typically
- In project, similarly,
MCq  1000  C q
(Marginal Cost in dollars per drive)
-
Derivatives
• Project (Marginal Cost)
- Calculate MC(q)
Nested If function, the if function using values for
Q1-4 & 6
- IF(q<=800,160,IF(q<=1200,128,72))
In the GOLDEN sheet need to use cell referencing
for IF function because we will make copies of
it, and do other project questions
=IF(B30<$E$20,$D$20,IF(B30<$E$22,$D$21,$D$22))
Recall -Production cost estimates
•
•
Fixed overhead cost - $ 135,000,000
Variable cost (Used for the MC(q)
function)
1) First 800,000 - $ 160 per drive
2) Next 400,000- $ 128 per drive
3) All drives after the first 1,200,000$ 72 per drive
Derivatives
• Project (Marginal Profit)
MP(q) = MR(q) – MC(q)
- If MP(q) > 0, profit is increasing
- If MR(q) > MC(q), profit is increasing
- If MP(q) < 0, profit is decreasing
- If MR(q) < MC(q), profit is decreasing
Derivatives
• Project (Maximum Profit)
- Maximum profit occurs when MP(q) = 0
- Max profit occurs when MR(q) = MC(q) & MP(q) changes
from positive to negative
- Estimate quantity from graph of Profit
- Estimate quantity from graph of Marginal Profit
Derivatives
• Project (Answering Questions 1-3)
1. What price? $285.88
2. What quantity? 1262(K’s) units
3. What profit? $42.17 million
Derivatives
• Project (What to do)
- Create one graph showing MR and MC
- Create one graph showing MP
- Prepare computational cells answering your
team’s questions 1- 3
Marginal AnalysisR ( q  h)  R ( q  h)
2h
 MR(q) = R′(q) ∙ 1,000
 R(q) 


where h = 0.000001
0.160 if 0  q  800

C (q)  0.128 if 800  q  1, 200
0.072 if q  1, 200

160 if 0  q  800

MC (q)  C (q) 1, 000  128 if 800  q  1, 200
 72 if q  1, 200

Marketing Project
Marginal Analysis R(q) 
R ( q  h)  R ( q  h)
2h
where h = 0.000001
 In Excel we use derivative of R(q)
 R(q)=aq^3+bq^2+cq
 R’(q)=a*3*q^2+b*2*q+c

Marketing Project
Marginal Analysis (continued)Marginal Revenue and Cost
$600
MR(q)
$ Per Drive
$400
MC(q)
$200
$0
0
400
800
1,200
1,600
2,000
2,400
2,800
-$200
-$400
-$600
q (K's)
Marketing Project
Marginal Analysis
 MP(q) = MR(q) – MC(q)
Marginal Profit
MP(q) $ Per Drive
$400
$200
$0
0
400
800
1,200
1,600
2,000
-$200
-$400
q (K's)
 We will use Solver to find the exact value of q for
which MP(q) = 0. Here we estimate from the graph
Marketing Project
Profit Function
 The profit function, P(q), gives the relationship
between the profit and quantity produced and sold.
 P(q) = R(q) – C(q)
P (q ) (M's)
Profit Function
$70
$60
$50
$40
$30
$20
$10
$0
-$10 0
-$20
400
800
1,200
q (K's)
1,600
2,000
Goals
•
1. What price should Card Tech put on the drives,
in order to achieve the maximum profit?
•
2. How many drives might they expect to sell at
the optimal price?
•
3. What maximum profit can be expected from
sales of the 12-GB?
•
4. How sensitive is profit to changes from the
optimal quantity of drives, as found in Question 2?
•
5. What is the consumer surplus if profit is
maximized?
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Goals-Contd.
•
6. What profit could Card Tech expect, if they price the
drives at $299.99?
•
7. How much should Card Tech pay for an advertising
campaign that would increase demand for the 12-GB drives by 10%
at all price levels?
•
8. How would the 10% increase in demand effect the
optimal price of the drives?
•
9. Would it be wise for Card Tech to put $15,000,000 into
training and streamlining which would reduce the variable production
costs by 7% for the coming year?
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Reminder
In HW 4 problems- methods of marginal analysis
(except project 1 focus problems)
Rq  h   Rq 
R' (q)  MR q  
h
C q  h   C q 
C ' (q)  MC q  
h
Pq  h   Pq 
P' (q)  MP q  
h