P13.6 SOLUTION

A. Because MCA = 0, Firm A’s profit-maximizing output level is found by setting MRA

= MCA:

MRA = MCA

$1,250 - $2QA - QB = 50

$2QA = $1,200 - QB

QA = 600 - 0.5QB

Notice that the profit-maximizing level of output for Firm A depends upon the level

of output produced by itself and Firm B. Similarly, the profit-maximizing level of

output for Firm B depends upon the level of output produced by itself and Firm A.

These relationships are each competitor’s output-reaction curve

Firm A output-reaction curve: QA = 600 - 0.5QB

Firm B output-reaction curve: QB = 600 - 0.5QA

B. The Cournot market equilibrium level of output is found by simultaneously solving

the output-reaction curves for both competitors. To find the amount of output

produced by Firm A, simply insert the amount of output produced by competitor

Firm B into Firm A’s output-reaction curve and solve for QA. To find the amount of

output produced by Firm B, simply insert the amount of output produced by

competitor Firm A into Firm B’s output-reaction curve and solve for QB. For

example, from the Firm A output-reaction curve

QA = 600 - 0.5QB

QA = 600 - 0.5(600 - 0.5QA)

QA = 600 - 300 + 0.25QA

0.75QA = 300

QA = 400 (000) units

Similarly, from the Firm B output-reaction curve, the profit-maximizing level of

output for Firm B is QB = 400. With just two competitors, the market equilibrium

level of output is

Cournot equilibrium output = QA + QB

= 400 + 400

= 800 (000) units

The Cournot market equilibrium price is $1,250 – Q = $1,250 - $1(800) = $450

P13.7 SOLUTION

A. To illustrate Stackelberg first-mover advantages, reconsider the Cournot model but

now assume that Firm A, as a leading firm, correctly anticipates the output reaction

of Firm B, the following firm. With prior knowledge of Firm B’s output-reaction

curve, QB = 600 - 0.5QA, Firm A’s total revenue curve becomes TRA = $1,250QA

- QA2 - QAQB

= $1,250QA - QA2 - QA(600 - 0.5QA)

= $650QA - 0.5QA2

With prior knowledge of Firm B’s output-reaction curve, marginal revenue for Firm

A is MRA = ∂TRA/∂QA = $650 - $1QA

Because MCA = $50, Firm A’s profit-maximizing output level with prior knowledge

of Firm B’s output-reaction curve is found by setting MRA = MCA = $50:

MRA = MCA

$650 - $1QA = $50

QA = 600

After Firm A has determined its level of output, the amount produced by Firm B is

calculated from Firm B’s output-reaction curve

QB = 600 - 0.5QA

= 600 - 0.5(600)

= 300

With just two competitors, the Stackelberg market equilibrium level of output is 900

and price is $350.

Notice that market output is greater in Stackelberg equilibrium than in Cournot

equilibrium because the first mover, Firm A, produces more output while the

follower, Firm B, produces less output. Stackelberg equilibrium also results in a

lower market price than that observed in Cournot equilibrium. In this example, Firm

A enjoys a significant first-mover advantage. Firm A will produce twice as much

output and earn twice as much profit as Firm B so long as Firm B accepts the output

decisions of Firm A as given and does not initiate a price war. If Firm A and Firm B

cannot agree on which firm is the leader and which firm is the follower, a price war

can break out with the potential to severely undermine the profitability of both

leading and following firms. If neither duopoly firm is willing to allow its

competitor to exercise a market leadership position, vigorous price competition and a

competitive market price/output solution can result. Obviously, participants in

oligopoly markets have strong incentives to resolve the uncertainty surrounding the

likely competitor response to leading-firm output decisions.

P13.8 SOLUTION

A. To derive Coke’s optimal price-response curve, set

∂πC/∂PC = 0

15 - 5PC + 1.25PP +2.5X = 0

5PC = 15 + 1.25PP + 2.5X

PC = $3 + $0.25PP + $0.5X

Coke’s optimal price-response curve shows that Coke should increase its own price

by 25¢ with each $1 increase in the price of Pepsi, and increase its own price by 50¢

with every $1 increase in the marginal cost of production.

B. If Pepsi charges $5 and marginal costs are $2 per 24-pack, Coke’s optimal price-

response curve shows that Coke should charge $5.25 per 24-pack:

PC = $3 + $0.25PP + $0.5X

= $3 + $0.25($5) + $0.5($2)

= $5.25

P14.3 SOLUTION

A. Yes, the dominant strategy for firm A is “up.” Notice that if firm B chooses “left,”

the highest payoff of $5 million can be achieved if Firm A chooses “up.” On the

other hand, if firm B chooses “right,” the highest payoff of $7.5 million can be

achieved if firm A again chooses “up.” No matter what firm B chooses, the highest

payoff results for firm A occurs if A chooses “up.” Therefore, “up” is a dominant

strategy for firm A.

B. No, there is no dominant strategy for firm B. If firm A chooses “up,” the highest

payoff of $10 million can be achieved if firm B chooses “left.” On the other hand, if

firm A chooses “down” the highest payoff of $5 million can be achieved if firm B

chooses “right.” Therefore, there is no dominant strategy for firm B. The profit-

maximizing choice by firm B depends upon the choice made by firm A.

P14.4 SOLUTION

A. Yes, the secure strategy for The Home Depot is to offer 90-day free financing.

Irrespective of the choice made by the Lowes Companies, in its secure strategy The

Home Depot can insure that it avoids the worst-possible outcome of earning only $15

million by choosing to offer 90-day free financing.

B. Yes, the secure strategy for the Lowes Companies is to offer 90-day free financing.

Irrespective of the choice made by The Home Depot, Lowes’ secure strategy insures

that it avoids the worst-possible outcome of earning only $10 million by choosing to

offer 90-day free financing.

P14.5 SOLUTION

A. A set of strategies constitutes a Nash equilibrium if no player can improve their

payoff through a unilateral change in strategy. The concept of Nash equilibrium is

important because it represents a stable situation in which no player can improve

their situation given the strategies adopted by other players.

B. Yes. The Nash equilibrium strategy is for both Kellogg and General Mills to

advertise. Given that Kellogg chooses to advertise, General Mills makes the most

profit by also choosing to advertise. Similarly, given that General Mills has chosen

to advertise, the best Kellogg can do is to advertise as well. Given the dual decision

to advertise, neither competitor can improve profits by changing its advertising

decision.

P14.6 SOLUTION

A. In this problem, the low-price strategy is a dominant strategy for both firms. If firm

B charged low prices, firm A will also choose to charge low prices because the $5

million profit then earned is more than the $10 million loss that would be suffered by

firm A if it pursued a high-price strategy. If firm B charged high prices, firm A

would still choose to charge low prices because the $40 million profit then earned is

more than the $25 million profit that would be earned if firm A pursued a high-price

strategy. If firm A charged low prices, firm B will also choose to charge low prices

because the $5 million profit then earned is more than the $10 million loss that would

be suffered by firm B if it pursued a high-price strategy. If firm A charged high

prices, firm B would still choose to charge low prices because the $40 million profit

then earned is more than the $25 million profit that would be earned if firm B

pursued a high-price strategy.

In this case, if both firms pursue a low-price strategy a Nash equilibrium also

results. A set of strategies constitutes a Nash equilibrium if no player can improve

their payoff through a unilateral change in strategy. The concept of Nash equilibrium

is important because it represents a stable situation in which no player can improve

their situation given the strategies adopted by other players.

B. If the firms agreed to collude and charge high prices, both would earn $25 million

and joint profits of $50 million would be maximized. However, the joint high-price

strategy is not a stable equilibrium. To see the instability of having both firms

choose high-price strategies, see how each firm has strong incentives to cheat on any

covert or overt agreement to collude. If firm B chose a high-price strategy, firm A

could see profits jump from $25 million to $40 million by switching from a high-

price to a low-price strategy. Similarly, if firm A chose a high-price strategy, firm B

could see profits jump from $25 million to $40 million by switching from a high-

price to a low-price strategy. Both firms have strong incentives to cheat on any

covert or overt agreement for both of them to charge high prices. Such situations are

common and help explain the difficulty of maintaining cartel-like agreements.

P15.3 SOLUTION

A. εP = pricein change Percentage

outputin change Percentage

= 0.03-

0.15

= - 5

B. Given the solution to part (a), and using the optimal mark-up on cost formula

(Hirschey, p. 588), the optimal mark-up is 25%. Given a marginal cost of $120, and

using the optimal pricing formula (Hirschey’s “Pricing Rule-of-Thumb”, p. 586), the

optimal price is $150.

P15.4 SOLUTION

A. EP = Q + Q

P + P

P

Q

12

12

= 250 + 750

$16 + $12

$16 - $12

250 - 750

= -3.5

B. Given εP = EP = -3.5, the optimal markup on cost for Saturday brunch at the Bristol

during this time frame is:

Coston Markup Optimal

= 1+

1-

P

= 1+3.5-

1-

= 0.4 or 40%

Given MC = $8.56, the optimal price is:

Coston Markup Optimal

= MC

MC - P

0.4 = $8.56

$8.56 - P

$3.424 = P - $8.56

P = $11.99

P15.5 SOLUTION

A. The $3 price increase to $39 represents a moderate 7.7 percent rise in price. Using

the arc price elasticity formula, the implied arc price elasticity of demand for Betty's

blouses is:

B. If it can be assumed that this arc price elasticity of demand EP = -2 is the best

available estimate of the current point price elasticity of demand, the optimal markup

on cost is:

Betty's standard cost per blouse includes the $12 purchase cost, plus $6 allocated

variable costs, plus $6 fixed overhead charges. However, for pricing purposes, only

the $12 purchase cost plus the allocated variable overhead charge of $6 are relevant.

Thus, the relevant marginal cost for pricing purposes is $18 per blouse. The

allocated fixed overhead charge of $6 is irrelevant for pricing purposes because fixed

overhead costs are unaffected by blouse sales.

At the $36 price, Betty's actual markup on relevant marginal costs per blouse is an

optimal 100 percent, because

Markup on Cost = $18

$18 - $36

= 1 (or 100 percent).

Therefore, Betty's initial $36 price on blouses is optimal, and the subsequent $3 price

increase should be rescinded.

2.- =

54 + 46

$36 + $39

$36 - $39

54 - 46 =

Q + Q

P + P

P - P

Q - Q = E

12

12

12

12P

100%.or 1 =

1 + 2-

1- =

1 +

1- =

Coston Markup Optimal

P

P15.7 SOLUTION

A. The incremental net income from these offers can be determined as follows:

Offer 1

Offer 2

Unit price

$14.60

$14.00

Unit variable costs:

Materials

$6.00

$6.00

Direct labor

4.00

4.00

Variable indirect labor

2.00

2.00

Variable warranty expense

1.20

13.20

0.00

12.00

Unit incremental profit

1.40

2.00

Units to be sold

× 80,000

×120,000

Total variable profit on units sold at special price

$112,000

$240,000

Less variable profit lost on regular sales:

Regular price

$20.00

Regular variable costs

- 13.20

Regular variable profit

6.80

Units that cannot be sold at regular price if

Offer 2 is accepted

×20,000

Opportunity cost of lost regular sales

$0

$136,000

Incremental profit

$112,000

$104,000

Both offers involve a substantial incremental profit, but offer 1 appears to be the

more attractive on a simple dollar basis.

But wait … Doesn’t this mean they will sell below cost once we factor in the $3 for

fixed overhead? To understand the fault of this logic ask yourself this: “Is that $3

AFC still an operative assumption once output expands to 480,000 or 500,000?”

What are total fixed costs anyway?

B. (i) The image of GE’s quality may be affected by sales of the appliance in

the department store chain with a private label.

(ii) Other buyers may demand the reduced price if GE accepts offer 1 and

the department store undercuts them at the retail price level.

(iii) The sales lost if GE accepts offer 2 may affect future orders from

regular customers.

C. It depends upon how you evaluate the factors discussed in part B. The incremental

profits of offer 1 exceed those of offer 2, but other long-run concerns might well

dictate that it not be accepted.

P15.8 SOLUTION

A. With price discrimination, profits are maximized by setting MR = MC in each

market, where MC = $10,000.

Wholesale

MRW = MC

$15,000 - $10QW = $10,000

QW = 500 units

PW = $12,500.

Retail

MRR = MC

$50,000 - $40QR = $10,000

QR = 1,000 units

PR = $30,000

The profit contribution earned by the company is:

π = PWQW + PRQR - AVC(QW + QR)

= $12,500(500) + $30,000(1,000)

- $10,000(500 + 1,000)

= $21,250,000

B. Yes, the point price elasticity of demand for each customer class is:

Wholesale

QW = 3,000 - 0.2PW

εP = ∂QW/∂PW × PW/QW

= -0.2 × ($12,500/500)

= -5

Retail

QR = 2,500 - 0.05PR

εP = ∂QR/∂PR × PR/QR

= -0.05 × ($30,000/1,000)

= -1.5

A higher price for retail customers is consistent with the lower degree or price

elasticity observed in that market.

P15.10 SOLUTION

A. It is appropriate to begin analysis of this problem by examining the optimal activity

level, assuming the firm mines and sells equal quantities of silver and lead.

For profit maximization where Q = QS = QL, set:

MC = MRS + MRL = MR

$10 = $11 - $0.00006Q + $0.4 - $0.00001Q

$0.00007Q = 1.4

Q = 20,000

Profit maximization with equal sales of each product requires that the firm mine Q =

20,000 tons of ore. Under this assumption, marginal revenues for the two products

are:

MRS = $11 - $0.00006(20,000) = $9.80

MRL = $0.4 - $0.00001(20,000) = $0.20

Because each product is making a positive contribution to marginal costs of $10 per

ton, Q = 20,000 is an optimal activity level.

Relevant prices are:

PS = $11 - $0.00003(20,000) = $10.40

PL = $0.4 - $0.000005(20,000) = $0.30

B. A five-fold (or 500%) increase in silver demand means that a given quantity could be

sold at 5 times the original price. Alternatively, 5 times the original quantity

demanded could be sold at a given price. Therefore, the new silver demand and

marginal revenue curves can be written:

PS’ = 5($11 - $0.00003QS)

= $55 - $0.00015QS

MRS’ = 5($11 - $0.00006QS)

= $55 - $0.0003QS

Now, assuming all output is sold,

MC = MRS’ + MRL = MR

$10 = $55 - $0.0003Q + $0.4 - $0.00001Q

0.00031Q = 45.4

Q = 146,452

Thus, profit maximization with equal sales of each product requires that the firm

mine Q = 146,452 tons of ore. Under this assumption, marginal revenues for the two

products are:

MRS’ = $55 - $0.0003(146,452) = $11.06

MRL = $0.4 - $0.00001(146,452) = -$1.06

Even though MRS’ + MRL = MC = $10, the above Q = 146,452 solution is

suboptimal. MRS’ = $11.06 > $10 = MC implies that a $1.06 profit contribution was

earned on each marginal ton of ore mined when just considering S sales. This means

that the firm would like to expand production beyond Q = 146,452 just to sell more

S. The negative marginal revenue for L implies that the firm had to reduce price so

much in order to sell all 146,452 pounds of L (indeed offer a negative price or

subsidy of 33¢ per pound) that total revenues fell by $1.06 on the last pound sold.

Rather than sell L under such unfavorable conditions, the firm would like to reduce L

sales below 146,452 pounds.

The firm would sell L only up to the point where MRL = 0 because, given additional

production to sell S, the marginal cost of L is zero. Set,

MRL = MCL

$0.4 - $0.00001Q = 0

$0.00001Q = 0.4

QL = 40,000

PL = $0.4 - $0.000005(40,000)

= $0.20

The optimal production and sales level of S is found by setting MRS = MC, because

S is the only product sold from the marginal ton of ore being mined.

MRS = MC = MCS

$55 - $0.0003QS = $10

0.0003QS = 45

QS = 150,000

and

PS = $55 - $0.00015(150,000)

= $32.50

Therefore, the firm should mine 150,000 tons of ore, and sell all 150,000 ounces of S

produced at a price of $32.50. Only 40,000 pounds of lead should be sold at a price

of 20¢ per pound, with the remaining 110,000 pounds produced being held off the

market.

(Note: Despite a five-fold increase in demand, prices increase by less than five-fold

given the firm’s expansion in output.)

Extra Problems

1. a. Firm 1’s reaction function is q1 = 60 – 0.5q2.

b. The competitive output is 120. By applying known relationships among the outcomes of

the various market structures, the rest of the table may quickly be filled in. For example,

the monopoly output will be ½ of the competitive output, and so forth.

2. a. No.

b. The Nash Equilibrium {6,6} is a prisoner’s dilemma.

c. The relevant portion of the game matrix is symmetric. Thus the same critical discount

factor will apply to both players. It is (20 – 15)/(21 – 20) = 5.

d. Equilibrium is {8,4}. Notice that this outcome incorporates the notion of a first mover

advantage and that the outcome mirrors that of a Stackelberg equilibrium.

3. a. = - 3

b. Optimal markup on cost is 50%. (Use the optimal markup formula, Hirschey, p. 588.)

Optimal price is $45. (Use Hirschey’s “Rule-of-Thumb” from p. 586.)

4. a. With price discrimination, profits are maximized by setting MR = MC in each market, where MC

= AVC = $20 (because AVC is constant).

Locals

MRL = MC

$40 $0.001QL = $20

0.001QL = 20

QL = 20,000 and PL = $30

Tourists

MRT = MC

$50 $0.0008QT = $20

0.0008QT = 30

QT = 37,500 and PT = $35

The profit contribution earned by the Fun-Land Amusement Park is $762,500.

b. Yes, a higher price for Tourist customers is consistent with the lower degree of price elasticity

observed in that market. Evaluating at equilibrium outputs and prices, elasticity of demand for

locals is – 3, and elasticity of demand for tourists is -2.33.