View
212
Download
0
Category
Tags:
Preview:
Citation preview
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
•
qCqCqMC 1
Dinner Example
Example 1. We consider the cost function C(q) = C0 + VC(q) = $63,929.37 + 13,581.51ln(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
)ln()1ln(51.581,13
)ln(51.581,1337.929,63)1ln(51.581,1337.929,63
)()1()(
qCqCqMC
We can use a calculator or Excel to compute values of MC(q). For
example,
.78906.6
)000,2ln()001,2ln(51.581,13
)000,2ln()1000,2ln(51.581,13)000,2(
MC
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.
(material continues)
Differentiation. Differentiation. Marginal Analysis: page 2Marginal Analysis: page 2Differentiation. Differentiation. Marginal Analysis: page 2Marginal Analysis: page 2
ITDinners.xls C
Marginal Cost Function, First Plan
$0
$4
$8
$12
$16
0 1,000 2,000 3,000 4,000
q dinners
MC
(q)
$ /d
inne
rDifferentiation,
MarginalDifferentiation. Differentiation. Marginal Analysis: page 3Marginal Analysis: page 3Differentiation. Differentiation. Marginal Analysis: page 3Marginal Analysis: page 3
(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.
IT CDinners.xls
Marginal Analysis- MC(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
•
• Typically use with h = 0.001
2lim
0 h
hqChqCqMC
h
2h
hqChqCqMC
Marginal Analysis
• Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). h=0.001
•
607.01278000 qqC
unitper 71.6$002.0
76585.1352177926.13521002.0
999.4991278000001.5001278000
001.02
999.499001.500500
607.0607.0
CC
MC
In terms of money, the marginal cost at the production level of 500, $6.71 per unit
C(q) = $63,929.37 + 13,581.51ln(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
•
h
hqPhqPqMP
h
hqChqCqMC
h
hqRhqRqMR
h
h
h
2lim
2lim
2lim
0
0
0
Differentiation, Marginal
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).
Demand Function
$0
$8
$16
$24
$32
0 1000 2000 3000 4000q
D(q
)
Marginal Demand Function
-$0.020
-$0.015
-$0.010
-$0.005
$0.000
0 1,000 2,000 3,000 4,000
q
MD
(q)
$/di
nner
Differentiation. Differentiation. Marginal Analysis: page 8Marginal Analysis: page 8Differentiation. Differentiation. Marginal Analysis: page 8Marginal Analysis: page 8
(material continues) I
Values for all of our mar-ginal 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
T CDinners.xls
Differentiation, Marginal
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.
Where the revenue function R(q) is increasing, the difference quotients for marginal revenue are
(material continues)
Revenue Function
$0
$10,000
$20,000
$30,000
$40,000
$50,000
0 1000 2000 3000 4000q
R(q
)
Marginal Revnue Function
-$40
-$20
$0
$20
$40
0 1,000 2,000 3,000 4,000
q
MR
(q)
$/di
nner
Differentiation. Differentiation. Marginal Analysis: page 9Marginal Analysis: page 9Differentiation. Differentiation. Marginal Analysis: page 9Marginal Analysis: page 9
IT CDinners.xls
Revenue and Cost Function
$0$10,000
$20,000$30,000$40,000
$50,000$60,000
0 1000 2000 3000 4000q
Dol
lars
Revenue
Cost
Marginal Revenue & Marginal Cost Functions
-$30
-$20
-$10
$0
$10
$20
$30
0 1,000 2,000 3,000 4,000
q
$/di
nner
M Revenue
M Cost
Profit Function
-$6,000
-$4,000
-$2,000
$0
$2,000
$4,000
$6,000
0 1000 2000 3000 4000
q
P(q
)
Differentiation, Marginal
Marginal analysis can tell us a great deal about the profit function. Refer back to these plots while reading the next pages.
Differentiation. Differentiation. Marginal Analysis: page 10Marginal Analysis: page 10Differentiation. Differentiation. Marginal Analysis: page 10Marginal Analysis: page 10
(material continues) IT CDinners.xls
Derivatives
• Project (Marginal Revenue)
- Typically
- In project,
-
qRqMR
qRqMR 1000
15
Recall:Revenue function-R(q)
• Revenue in million dollars R(q)
• Why do this conversion?Marginal Revenue in dollars per drive
qRqMRtypically
qR
qRqMR
10001000
1000000
Derivatives
• Project (Marginal Cost)
- Typically
- In project, similarly,(Marginal Cost in dollars per drive)
-
qCqMC
qCqMC 1000
Derivatives
• Project (Marginal Cost)- Calculate MC(q)
Nested If function, the if function using values for Q1-4 & 6- IF(q<=800,160,IF(q<=1200,128,72))
In the GOLDEN sheet need to use cell referencing for IF function because we will make copies of it, and do other project questions
=IF(B30<$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 Analysis-
where h = 0.000001
MR(q) = R′(q) ∙ 1,000
hhqRhqR
qR2
)()()(
0.160 if 0 800
( ) 0.128 if 800 1,200
0.072 if 1,200
q
C q q
q
160 if 0 800
( ) ( ) 1,000 128 if 800 1,200
72 if 1, 200
q
MC q C q q
q
Marketing Project
Marginal Analysis-
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
hhqRhqR
qR2
)()()(
Marketing Project
Marginal Analysis (continued)-
Marketing Project
Marginal Analysis
MP(q) = MR(q) – MC(q)
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)
Profit Function
-$20-$10
$0$10$20$30$40$50$60$70
0 400 800 1,200 1,600 2,000
q (K's)
P(q
) (M
's)
28
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?
29
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?
ReminderIn HW 4 problems- methods of marginal analysis
(except project 1 focus problems)
h
qPhqPqMPqP
h
qChqCqMCqC
h
qRhqRqMRqR
)('
)('
)('
Recommended