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

EX:

Recall Dinner problem

C(q) = C0 + VC(q). = 9000+177*q0.633

MC(q)- the marginal cost at q dinners

MC(100)- gives us the marginal cost at 100 dinners

This means the cost per unit at 100 dinners

How to find MC(q) ? We will learn 3 plans

Marginal Analysis

• First Plan

• Cost of one more unit

•

qCqCqMC 1

Marginal Analysis

• Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

•

607.01278000 qqC

unitper 70.6$

77255.1352147335.13528

50012780005011278000

500501500607.0607.0

CCMC

Marginal Analysis

• Second Plan

• Average cost of one more and one less unit

•

2

11

qCqCqMC

2

11 from derived

qCqCqCqC

Marginal Analysis

• Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

•

607.01278000 qqC

unitper 70.6$2

06648.1351547335.135282

49912780005011278000

2

499501500

607.0607.0

CCMC

Marginal Analysis

• 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).

•

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

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

Marginal Analysis

• Graphs

29303132333435

Demand Marginal Demand

D(q) is always decreasingAll the difference quotients for marginal demand are negativeMD(q) is always negative

Marginal Analysis

• Graphs

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Revenue Marginal Revenue

R(q) is increasing • difference quotients for marginal revenue are positive•MR(q) is positive

R(q) is decreasing• difference quotients for marginal revenue are negative•MR(q) is negative

Maximum revenue

Marginal revenue 0

q1

q1

Marginal Analysis

• Graphs

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Profit Marginal Profit

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Cost Marginal Cost

Derivatives-part 2

• Difference quotients

•

• Called the derivative of f(x)

h

hxfhxfxfm

h 2lim

0

Derivatives

• Ex1. Evaluate if

62 xxf

5452.5002.0

0110903561.0002.0

99445674.130055471.14001.02

62623

999.2001.3

f

3f

Derivatives

• Differentiating.xls file

• Graphs both function and derivative

• Can evaluate function and derivative

Derivatives

• Differentiating.xlsIncrement

x f (x ) f ' (x ) a b h s

= = #VALUE! 0.000001 t

u

v

w

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ConstantsDefinition Plot Interval

Formula for f (x )

Computation

FUNCTION

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x

f (x )

DERIVATIVE

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x

f ' (x )

Derivatives

• Use Differentiating.xls to graph the derivative of on the interval [-2, 8]. Then

evaluate . 62 xxf

3f DERIVATIVE

0

50

100

150

200

-5 0 5 10

x

f ' (x )

5452.53 f

62 xxf

Differentiating.xslm

Incrementx f (x ) f ' (x ) a b h

7 3 14 5.545 -2 8 0.000001

Definition Plot IntervalFormula for f (x )

Computation

Key points1)Formula for the function in x2)Plot Interval is ESSENTIAL3)You can use the computation cells to evaluate the f(x) & f’(x) at different values4)If using Office 2007, save it as a macro enabled file

Derivatives

• PropertiesIf (c is a constant) then

If (m is a constant) then

If then

If then

xfaxg xfaxg

xgxfxh xgxfxh

cxf 0 xf

mxxf mxf

Derivatives

• Tangent line approximations

• Useful for easy approximations to complicated functions

• Need a point and a slope (a derivative)

• Use y = mx +b

Derivatives

• Ex. Determine the equation of the tangent line to at x = 3.

• Recall and we have the point (3, 14)

• Tangent line is y = 5.5452x – 2.6356

62 xxf

5452.53 f

Derivatives

• Project (Marginal Revenue)

- Typically

- In project,

- units are

qRqMR

qRqMR 1000

h

hqRhqRqR

2

units thousand

dollarsmillion

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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,

- units are

qCqMC

qCqMC 1000

h

hqChqCqC

2

units thousand

dollarsmillion

Derivatives

• Project (Marginal Cost)

- Marginal Cost is given in original data

- Cost per unit at different production levels

- Use IF function in Excel

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 (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 (Maximum Profit)

- Maximum profit occurs when MP(q) = 0

- Max profit occurs when MR(q) = MC(q)

- Estimate quantity from graph of Profit

- Estimate quantity from graph of Marginal Profit

Derivatives-change

• 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)/1000

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)

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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?

39

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?