Differentiation Purpose- to determine instantaneous rate of change Eg: instantaneous rate of change...

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()(

qq

qq

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


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


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


(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


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


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


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

)


Marginal analysis can tell us a great deal about the profit function. Refer back to these plots while reading the next pages.


(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

)('

)('

)('

Differentiation Purpose- to determine instantaneous rate of change Eg: instantaneous rate of change...

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