engineering economics monopoly numerics

7/30/2019 Engineering Economics Monopoly Numerics

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Pricing

Strategies


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Some pricing strategies that we will explore:

1. Price discrimination 1st, 2nd, & 3rd degree

2. Two-part Tariff pricing

3. Bundling

4. Advertising

5. Cost-plus markup pricing

6. Product Lines

7. Peak-Load pricing

8. Transfer Pricing


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Charging different prices to different

consumers for the same product. Enables firms to charge some consumers

higher prices, and to capture consumersurplus.

Price Discrimination


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Under what conditions isprice discrimination possible?

a. Firm must have some control over price.

b. The firm can identify different submarkets.

c. The submarkets have different priceelasticities of demand.

d. The firm can prevent arbitrage(the purchase of an item for immediate resale

in order to profit from the price discrepancy).


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3 Types of Price Discrimination

First Degree

Each customer is charged the maximum price thatthey are willing to pay.

Second Degree involves self-selectionOne type of 2nd degree price discrimination is blockpricing or quantity discounts in which firms chargedifferent prices depending on volume of usage.

Third Degree or multi-market (most common)

Markets distinguished by other factors.


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Since under first degree price discrimination,each customer is charged the maximumprice that they are willing to pay, consumersurplus is zero.


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First Degree Price Discrimination

Note: Each time the firmsells another unit, itincreases its revenues bythe price for which it sellsthat unit.

Unlike the usual situation, itdoesnt need to lower toprice to all the othercustomers in order to sell tothe additional one.

So P=MR & the Demand &MR curves are the same,with 1st degree pricediscrimination.

Q

D

$= MR


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First degree price discriminationExample:

Suppose the demand curve for a monopolists product is:P = 9 0.005 Q.

The average total cost curve is a horizontal line:

ATC = MC = 1.5(1) Determine the price, quantity, consumer surplus,

producer surplus (profit), & the sum of consumer &producer surplus, if the firm does NOT pricediscriminate.

(2) Determine the quantity, consumer surplus, producersurplus (profit), & the sum of consumer & producersurplus, if the firm does price discriminate.


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First, if the firm does NOT price discriminate:

We have the demand curve for a monopolists product is:

P = 9 0.005 Q.& the average total cost curve is:

ATC = MC = 1.5

TR = PQ =(9 0.005 Q) Q = 9 Q 0.005 Q2 .

Then MR = dTR/dQ = 9 0.01 Q .

Setting MR = MC, we have 9 0.01Q = 1.5

or 7.5 = 0.01 Q .

So Q = 750,& P = 9 0.005 Q = 9 0.005 (750) = 5.25.

Profit or producer surplus = TR TC = PQ (ATC)Q= (5.25)(750) (1.5)(750) = 2812.5


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Without Price Discrimination

$

Q

D

P*= 5.25

Q*= 750

ATC =MC =1.5

Consumer Surplus

= (1/2)(750)(3.75)= 1406.25

MR

profit

9

Combined consumer & producer surplus is

CS + PS = 1406.25 + 2812.5 = 4218.75


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Now if the firm does1st degree price discrimination:

We had the demand curve: P = 9 0.005 Q,

& the average total cost curve: ATC = MC = 1.5

MR is now the same as the demand function,

so MR = 9 0.005 Q.

Setting MR = MC,

we have 9 0.005Q = 1.5

or 7.5 = 0.005 Q .

So Q = 1500.


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With 1st Degree Price Discrimination

$

Q

D=MR

Q*= 1500

ATC =MC =1.5

Profit= (1/2) (1500) (7.5)

= 5625

9

Combined consumer & producer surplus is

CS + PS = 0 + 5625 = 5625


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2nd Degree Price Discrimination: Block Pricing

Price is based on volume of usage of the good.

Those who consume large quantities are chargeda lower price.

Those consuming small quantities are charged ahigher price.


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Second Degree Price Discrimination Example

Suppose there are 100 high-volume consumers who value the 1st

unit of a good at $15 & a 2nd unit at $10.

There are also 100 low-volume consumers who value the 1st unitat $12.

The total cost of production is TC = 6 Q.(So ATC = TC/Q = 6Q/Q = 6.)

Determine the total revenue, total cost, producer surplus (profit),consumer surplus, & sum of the producer & consumer surplusfor the following four options:

1. No price discrimination one unit sells for $15.2. No price discriminationone unit sells for $12.

3. No price discriminationone unit sells for $10.

4. Offer two sizes of packages, 1 unit for $12 & 2 units for $20.


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High-volume consumersvalue the 1st unit of a good at$15 & the 2nd unit at $10.

$

Q1 2

15

10

$

Q1

12

Low-volume consumersvalue the 1st unit at $12.

ATC ATC6

6


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The 100 high-volumeconsumers will buy 1unit.

The 100 low-volumeconsumers will buy 0 units.

Suppose firm sells all units individually for $15.

TR = PQ = 15(100) = 1500,TC = 6 Q = 6 (100) = 600, &Producer Surplus or = TR TC = 1500 600 = 900.Consumer surplus = 0(100) = 0.

PS + CS = 900 + 0 = 900

$

Q1

2

15

10

ATC

$

Q1

12

ATC6

6

profit


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The 100 high-volumeconsumers will buy 1unit.

The 100 low-volumeconsumers will buy 1 unit.


TR = 12(200) = 2400,TC = 6 Q = 6(200) = 1200, &Producer Surplus or = TR TC= 2400 1200 = 1200.Consumer Surplus = 3(100) + 0(100) = 300

PS + CS = 1200 + 300 = 1500

$

Q1

2

15

10

ATC

$

Q1

12

ATC6 6

profit

12CS

profit


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The 100 high-volumeconsumers will buy 2 units.

The 100 low-volumeconsumers will buy 1 units.


TR = PQ = 10(300) = 3000,TC = 6 Q = 6 (300) = 1800, &Producer Surplus or = TR TC = 3000 1800 = 1200.Consumer surplus = 5(100) + 2(100) = 700.

PS + CS = 1200 + 700 = 1900

$

Q1

2

15

10

ATC

$

Q1

12

ATC66

10profitprofit profit

CSCS


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The high-volume consumerswill buy a 2-unit pack.

The low-volume consumerswill buy a 1-unit pack.

Suppose firm sells 1-unit packs for $12 & 2-unit packs for $20.

TR = PQ = 12(100) + 20(100)= 3200,TC = 6 Q = 6 (300) = 1800, &Producer Surplus or = TR TC = 3200 1800 = 1400.Consumer surplus = 5(100) + 0(100) = 500.

PS + CS = 1400 + 500 = 1900

$

Q1

2

15

10

ATC

$

Q1

12

ATC66

profit profit

CS


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In our 2nd degree price discrimination case, the firm offered twosizes of packages, 1 unit for $12 & 2 units for $20.

The 100 high-volume consumers value the 1st unit of a good at$15 & the 2nd unit at $10.

However, notice that if the firm tried to charge $25 for the 2-pack,

the high-volume consumers would only buy a 1-pack. This isbecause they would be better off with consumer surplus of$15 $12 = 3 with a 1-pack than consumer surplus of$25 $25 = 0 with a 2-pack.

The profit with 2nd order price discrimination is more than the

profit for the one-price options. PS+CS is the same as for theunit price of $10, but the producer has captured the $200 low-volume CS as PS or profit.


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Third Degree Price Discrimination

Charging different prices to different groups.

Example: Charging lower movie admissions tostudents & senior citizens than to other movie-goers.


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Third Degree Price Discrimination

For each group, the firm produces such thatMR = MC .

The group with the lowest elasticity pays the

highest price.

Example: Students & senior citizens mayhave more limited incomes, and therefore be

more responsive to changes in movie prices.Other movie-goers may be less responsive tochanges in movie prices.


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Suppose the demand functions for two groups of consumers areD1: P = 101 13Q and D2: P = 53 7 Q.

Notice that D1 is steeper and so less elastic than D2 .

(So group 1 will pay a higher price than group 2.)The total cost function is TC = 90 + 128Q 22Q2 + Q3 .

If the firm is able to price discriminate between the two groups,determine the prices that should be charged, the quantities that

will be purchased, total revenue, total cost, and profit.

We need to equate the two MR functions to the MC function.

MC = dTC/dQ = 128 44Q + 3Q2.

Group 1: TR1 = PQ = (101 13Q)Q = 101Q 13Q2 , and

MR1 = dTR1/dQ = 101 26 QGroup 2: TR2 = PQ = (53 7Q)Q = 53Q 7Q

2 , and

MR2 = dTR2/dQ = 53 14 Q


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Our Group 1 Demand function was P = 101 13 Q, andthe MR function was MR1 = 101 26 Q.

The MC function was MC = 128 44Q + 3Q2

.

Set MR1 = MC: 101 26 Q = 128 44Q + 3Q2

0 = 3Q2 18Q + 27

Dividing by 3 to simplify: 0 = Q2 6Q + 90 = (Q 3) (Q 3)

Q 3 = 0

So for Group 1, Q = 3

From Group 1s demand function, P1 = 101 13 (3) = 62.The revenue from Group 1 will be PQ = (62)(3) = 186.


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Our Group 2 Demand function was P = 53 7 Q, andthe MR function was MR2 = 53 14 Q.

The MC function was MC = 128 44Q + 3Q2.

Set MR2 = MC: 53 14 Q = 128 44Q + 3Q2

0 = 3Q2 30Q + 75

Dividing by 3 to simplify: 0 = Q2

10Q + 250 = (Q 5) (Q 5)

Q 5 = 0

So for Group 2, Q = 5

From Group 2s demand function, P2 = 53 7 (5) = 18.The revenue from Group 2 will be PQ = (18)(5) = 90


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Adding the revenues from the two groups together, weget TR = 186 + 90 = 276.

Since we produced 3 units for Group 1 and 5 forGroup 2, our production level is 8.

Plugging 8 into our total cost function,TC = 90 + 128Q 22Q2 + Q3 = 218.

So our profit is = TR TC = 276 218 = 58.


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The Two-Part Tariff

There are two components to the price: a unit price(P) for each unit consumed, & a tariff (T) for

entry into the market.Examples include BJs, telephone service, health

clubs, etc.

The tariff enables the firm to capture some

consumer surplus.


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Suppose that a firm has constant average and marginal

costs as shown.

P

D

ATC=MC

Q

P*

Q*

Also, each customer has the indicated

demand curve.Suppose that the firm charges price P* perunit.

Based on the per unit charge, the firm earnsrevenues equal to the area of the blue box.


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The firm can also pick up the consumer surplus,

P

D

ATC=MC

Q

P*

Q*

if it charges a membership feeequal to the area of the greentriangle.


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The firms total revenue (from each customer) is the

combined areas of the blue box and the green triangle.

P

D

ATC=MC

Q

P*

Q*

Recall that ATC = TC/Q.

So, TC = ATCQ.

So, the total cost (from eachcustomer) is the purple box.

The firms profit(per customer) isTR - TC which isthe orange figure.


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The firms profit from this two-part tariff strategywill be greatest if it produces where the Demand

and ATC = MC curves intersect, or P = ATC = MC.P

D

ATC=MC

Q

P*

Q*


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Two-Part Tariff Example

Suppose a firms TC function is TC = 5Q.

Suppose also that each of the firms customers

has this demand curve: P = 35 Q .

Determine the appropriate unit price andmembership fee for a two-part tariff pricingstrategy.

Also determine the quantity purchased, totalrevenue, total cost, and profit per customer.


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Demand function (for each person): P = 35 Q

Total cost function: TC = 5QAs we indicated previously, the firms profit will be

greatest if it produces where P = ATC = MC.

ATC = MC = 5

So, P = 35 Q = 5 ,

So, 30 = Q.

So revenue per person from per unit sales is

PQ = (5)(30) = 150 .Next we need to determine the appropriatemembership fee.


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P

D

ATC=MC

Q

5

30

The membership fee is the consumer surplus.

That is the area of the orange triangle,which is (1/2)(30)(30) = 450.

So the membership fee should be $450.35


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Combining the membership fee of $450 with the

per unit sales revenues of $150 that we foundearlier, we have total revenues per customer of$450 + $150 = $600.

From the total cost function, the total productioncost for the 30 units per customer isTC = 5Q = 5(30) = 150.

So our profit per customer is = TR TC = 600 150 = $450.


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Bundling

Bundling is packaging two or more products togain a pricing advantage.

Conditions necessary for bundling to be the

appropriate pricing alternative:Customers are heterogeneous.

Price discrimination is not possible.

Demands for the two products are negativelycorrelated.


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Consider the following reservations prices,for two buyers: Alan and Beth

Stereo TVSum of

reservationprices

Alan $225 $375 $600

Beth $325 $275 $600

Maximum price forboth to buy the good

$225 $275

To get both people to buy both goods without bundling, youcan only charge $225 + $275 = $500, & each person wouldhave consumer surplus of $600 $500 = $100.

If you bundle, you can charge $600 & consumer surplus = 0.


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The effectiveness of bundling as apricing strategy depends upon the

degree of negative correlation betweenthe demands for the two goods.


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The ratio of the firms advertising to its salesrevenue should equal the negative of the ratio

of the advertising & price elasticities ofdemand.

That is,A/(P*Q) = - A / D

So you should advertise a lot if the elasticity of

demand(1) with respect to advertising is high, &

(2) with respect to price is low.

Advertising:How does a firm determine

the profit-maximizing advertising level?


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Example: Suppose that elasticity of demand with

respect to advertising is 0.10, and elasticity ofdemand with respect to price is -0.50. Whatpercent of sales revenues should the advertisingbudget should be?

A/(P*Q) = - A / D = -0.10 / -0.50 = 0.20 or 20%


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Cost-Plus Pricing

The price charged by the firm is the average totalcost of production plus a percentage of that cost.

Example: If the average total cost of productionis $50, and the firm uses a 10% markup, the firm

will sell the product for $55.


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Pricing using Product Lines

A firm may have several lines of a product, such as

(1) a regular line,

(2) an economy product (for people who want to save money), &

(3) a top-of-the-line product (for people who want the best).

To maximize profit, the firm sets MR = MC for each product line.

P d t Li P i i E l A h 3 d t li


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Product-Line Pricing Example: A company has 3 product lines.deluxe: TC= 70 + 40Q + Q

2 & demand function is P = 90 4Qregular: TC = 65 + 30Q + Q2 & demand function is P = 84 2Qeconomy: TC = 50 + 20Q + Q2 & demand function is P = 60 Q

Determine the profit-maximizing price for each line.

For each product line, we want MR = MC. So for each line, we needto calculate MC = dTC/dQ, TR = PQ, & MR = dTR/dQ.

deluxe: MC = 40 + 2 QTR = (90 4Q)Q = 90 Q 4Q2MR = 90 8Q

regular: MC = 30 + 2 QTR = (84 2Q)Q = 84 Q 2Q2MR = 84 4Q

economy: MC = 20 + 2 QTR = (60 Q)Q = 60 Q Q2

MR = 60 2Q

F d t li i i l h f


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For our product-line pricing example, we have so far:deluxe: P = 90 4Q, MR = 90 8 Q MC = 40 + 2 Q,regular: P = 84 2Q, MR = 84 4Q MC = 30 + 2 Q,

economy: P = 60 Q, MR = 60 2Q MC = 20 + 2 Q,For each line we set MR = MC. So,

For the deluxe line,90 8 Q = 40 + 2 Q

50 = 10 Q & Q = 5.From the demand function, the deluxe price isP = 90 4 Q = 90 4(5) = 90 20 = 70.

For the regular line,84 4 Q = 30 + 2 Q54 = 6Q & Q = 9.The regular price is P = 84 2(9) = 84 18 = 66.

For the economy line,60 2 Q = 20 + 2 Q40 = 4 Q & Q = 10.The economy price is P = 60 10 = 60 10 = 50.

P k L d P i i


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Peak-Load PricingWhen demand is not evenly distributed, a firm needs to

have facilities to accommodate periods of highdemand.

Even with large facilities, the firm may experience timeswhen the demand is greater than can be handled.

Then the firm may experience costly computer systemcrashes.

During off-peak times (periods of lower demand), there isexcess capacity.

The firm charges less at off-peak times.

Example: More phone calls are made during businesshours than in the evenings and on weekends. So thephone companies charge more during business hours.

Peak Load Pricing Example:


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Peak-Load Pricing Example:Suppose the demand function for a firms service is

Peak times (days): P = 74 5 Q

Off-peak times (nights): P = 26 5 QThe marginal cost of providing the service is MC = 2 + 2Q .

Determine the day & night profit-maximizing prices.

We need to find when MR = MC for days & for nights.

For days,TR = PQ = (74 5 Q) Q = 74 Q 5 Q2

So MR = dTR/dQ = 74 10 Q .

MR = MC implies 74 10 Q= 2 + 2 Q,or 72 = 12 Q.

So Q = 6

& peak price is P = 74 5 Q = 74 5(6) = $44 per unit.

N t d t d th thi f i ht t fi d th


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Next we need to do the same thing for nights to find theoff-peak price.We had these demand functions:

Peak times (days): P = 74 5 QOff-peak times (nights): P = 26 5 Qand the marginal cost function was MC = 2 + 2Q .

For nights,

TR = PQ = (26 5 Q) Q = 26 Q 5 Q2

So MR = dTR/dQ = 26 10 Q .MR = MC implies 26 10 Q= 2 + 2 Q,

or 24 = 12 Q.So Q = 2

& off-peak price is P = 26 5 Q = 26 5(2) = $16 per unit(instead of $44 per unit as it was for peak times).


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Transfer Pricing

Sometimes firms are organized into separatedivisions.

One division may produce an intermediate

product and supply it to another division toproduce the final product.

How does the firm determine the efficient priceat which the intermediate product should besold. That is, what is the transfer price?


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The Simplest Case

The firm has 2 divisions: E and A

Division E produces the intermediate product (engine)for Division A which produces the final product(automobile).

Division E does not sell engines to anyone but divisionA, and division A does not buy engines from anyonebut division E.

Each unit of output (automobile) requires one unit ofthe input (engine).

The goal is to maximize the firms profit.


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First, find the companys (total) marginal cost MCT,which is the marginal cost of division Es producing anengine (MCE) plus the marginal cost of division Asproducing an auto (MCA).

That is, MCT = MCA + MCE .Then, produce the amount of output (autos) so that themarginal revenue from selling an auto (MR) is equal tothe marginal cost of production (MCT).

The appropriate price of the auto for that quantity ofoutput is determined from the demand curve for thefirms autos.

How do we determine the optimal quantity & pricefor the final product (the auto)?


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If the company determines the price of the engine,then division E is a price taker. So, PE and MRE will be

equal.The firm should set the price of the intermediateproduct (the engine) so that PE = MRE = MCE at theprofit-maximizing output level previously determined.

So what is the transfer price at which division E sellsthe intermediate product to division A?


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Transfer Pricing Example

A company has 2 divisions: production & marketing.

The production divisions total cost function isTCp = 70,000 + 15Q + 0.005 Q2.The marketing divisions total cost function is

TCm = 30,000 + 10 Q .The demand function for the final marketed product is

Pf= 100 0.001 Q .

What should be the price that transfers the product fromproduction to marketing?

Also determine the price of the final product and the firms profit.The marginal cost functions for the 2 divisions areProduction: MCp = dTCp/dQ = 15 + 0.01 Q

Marketing: MCm = dTCm/dQ = 10So thecombined MC = MCp + MCm = 25 + 0.01 QTR = PfQ = (100 0.001 Q) Q = 100 Q 0.001 Q

2

So MR = dTR/dQ = 100 0.002 Q

Continuing we have


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Continuing, we havedemand for the final product: Pf= 100 0.001 Q .TCp = 70,000 + 15Q + 0.005 Q

2 ; TCm = 30,000 + 10 Q .MCp = dTCp/dQ = 15 + 0.01 Q; MCm = dTCm/dQ = 10

MC = MCp + MCm = 25 + 0.01 Q ; MR = 100 0.002 Q

Equating MR & MC, we have

100 0.002 Q = 25 + 0.01 Q .

So 75 = 0.012 Q

& Q = 75/0.012 = 6,250 .

So the price of the intermediate product isPi = MCp = 15 + 0.01 (6,250) = $77.50 .

The price of the final product is

Pf= 100 0.001 Q = 100 0.001 (6,250) = $93.75 .Plugging the quantity 6,250 into the two total cost functions &adding, we find TC = TCp + TCm = $451,562.50 .

The total revenue is TR = PfQ = (93.75) (6250) = $585,937.50 .

So the firms profit is TR TC = $134,375 .

engineering economics monopoly numerics

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