Price and Output Determination: Oligopoly

Chapter 8 58

Chapter 8 Price and Output Determination: Oligopoly

Solutions to Exercises

1. a. A = PQA  TCA 2 = (200  QA QB)QA (1,500 + 55QA + QA ) 2 = 1,500 + 145QA  2QA  QAQB

A/QA = 145  4QA  QB = 0

2 B = 1,200 + 180QB  3QB  QAQB B/QB = 180  6QB  QA = 0

Solving these two equations simultaneously results in:

QB* = 25

QA* = 30 P* = $145/unit

2 b. A* = 1,500 + 145(30)  2(30)  30(25) = $300 2 B* = 1,200 + 180(25)  3(25)  30(25) = $675 * = 300 + 675 = $975

2.a. (Stackelberg) The profit- maximizing condition for Firm B is the same as before.

B/QB = 180  6QB  QA = 0

Firm A solves the above equation for QB in terms of QA to incorporate in its profit function Firm B’s reaction to quantity changes made by Firm A.

QB = 30 – QA /6 2 A = [200 – QA – (30 – QA /6)] QA – [1,500 + 55QA + QA ]

dA/dQA = MRA – MCA = (170 – 5QA /3) – (55 + 2QA) = 0

QA* = 345 / 11 = 31.36

QB* = 30 – 31.36 / 6 = 30 – 5.23 = 24.77 P* = 200 – 31.36 - 24.77 = $143.87

2 b. A = ($143.87) (31.36) - [1,500 + 55(31.36) + 31.36 ] 59 Chapter 8

A* = $4,511.76  $4,208.25 = $303.51 2 B = ($143.87) (24.77) – [1,200 + (20) (24.77) + (2) (24.77) ]

B* = $3,563.66  $2,922.51 = $641.15 * = $303.51 + $641.15 = $944.66

3.a. (Bertrand) Under Bertrand competition, each firm sets its MC = P. The two conditions are

200 – QA – QB = 55 + 2QA

200 – QA – QB = 20 + 4QB

The simultaneous solution to these equations is

QA* = 38.95, QB* = $ 28.21 P* = 200 – 38.95 - 28.21 = $132.84

2 b. A = ($132.84) (38.95) - [1,500 + 55(38.95) + (38.95) ]

A* = $5,174.12  $5,159.35 = $14.77 2 B = ($132.84) (28.21) – [1,200 + (20) (28.21) + (2) (28.21) ]

B* = $3,747.42  $3,355.81 = $391.61 * = $14.77 + $391.61 = $406.38

4. a.  = A + B 2 2 = 2700 + 145QA  2QA  2QAQB + 180QB  3QB

/QA = 145  4QA  2QB = 0

/QB = 180  6QB  2QA = 0

Solving these equations simultaneously yields:

QB* = 21.5

QA* = 25.5 p* = $153/unit

2 b. A* = 1500 + 145(25.5)  2(25.5)  25.5(21.5) = $348.75 2 B* = 1200 + 180(21.5)  3(21.5)  25.5(21.5) = $735 * = 348.75 + 735.00 = $1083.75

c. MCA = 55 + 2QA = 55 + 2(25.5) = $106/unit MCB = 20 + 4QB = 20 + 4(21.5) = $106/unit Oligopoly 60

Optimal Value Cournot Stackelberg Bertrand Form Cartel

QA* 30 31.36 38.95 25.50

QB* 25 24.77 28.21 21.50

QTotal*=QA*+QB* 55 56.13 67.16 47.00 P* $145 $143.87 $132.84 $153.00

A* $300 $303.51 $14.77 $348.75

B* $675 $641.15 $391.61 $735.00

Total*= A* + B* $975 $944.66 $406.38 $1083.75 a. Selling price is $8/unit higher than Cournot, $9.13 higher than Stackelberg, and $20.16 higher than Bertrand. b. Total industry output is 8 units lower than Cournot, 9.13 lower than Stackelberg, and 20.16 lower than Bertrand. c. Total industry profits are $108.75 higher than Cournot, $139.09 higher than Stackelberg, and $677.37 higher than Bertrand.

6. a. Alchem's (L) profit-maximizing output occurs where:

MRL = MCL

MRL is found as follows:

MRL = d(TRL)/dQL

TRL is given by the following expression:

TRL = P  QL

Also, QL is given by: QL = QT  QF

Using the total demand function, one can solve for QT: P = 20,000  4QT or QT = 5000  .25P

In order to find QF, one notes that Alchem lets the follower firms (F) sell as much polyglue as they wish at the given market price (P). Therefore, the follower firms are faced with a horizontal demand function and hence:

MRF = P

In order to maximize profits, the follower firms will operate where:

MRF = MCF , or where: 61 Chapter 8

P = 2000 + 4QF

Solving for QF yields:

QF = .25 P  500

Substituting QF and QT into the expression above for QL gives:

QL = (5000  .25P)  (.25P  500) = 5500  .50P Solving for P gives:

P = 11,000  2QL

Substituting P into the TRL expression above gives: 2 TRL = (11,000  2QL)QL = 11,000QL  2QL

MRL is therefore equal to: MRL = 11,000  4QL

Setting MRL = MCL gives: 11,000  4QL = 5000 + 5QL

QL* = 666.7

Substituting this value into the expression for P gives: P* = 11,000  2(666.7) = $9,666.7

b. From the expression above for QT, one obtains: QT* = 5000  0.25(9666.7) = 2,583.3

Also from the expression above for QF, one obtains:

QF* = 0.25(9666.7)  500 = 1,916.7

7. a. Segmented demand function:

P = 150  0.5Q, when 0  Q  50 200  1.5Q, for Q > 50 Oligopoly 62

Industry structure is oligopolistic. Firms have little or no incentive to raise prices since they cannot be sure competitors will follow. Similarly, firms have little or no incentive to lower prices since this will lead to retaliation by competitors. b. TR = P  Q = 150Q  .5Q2 when 0  Q  50 = 200Q  1.5Q2 when Q > 50 MR = 150  1.0Q when 0  Q  50 = 200  3Q when Q > 50

2 c. TC1 = 500 + 15Q + .5Q

MC1 = 15 + 1.0Q At the kink:

P1 = P2

where: P1 = 150 .5Q and P2 = 200  1.5Q 150  .5Q = 200  1.5Q Q* = 50 At Q* = 50:

MR1 = 150  1.0Q = 150  1.0(50) = 100

MR2 = 200  3.0(50) = 50

MC1 = 15 + 1.0(50) = 65

Profit-maximizing output occurs where:

MC1 = MR 63 Chapter 8

Since MR2 < MC1 < MR1, profit-maximizing output (Q*) is 50. Also, P* = 150  .5(50) = 125

2 d. TC2 = 500 + 45Q + .5Q MC2 = 45 + 1.0Q

At Q* = 50, MC2 = 45 + 50 = 95

Since MR2 < MC2 < MR1, profit-maximizing output (Q*) remains at 50. Also, P* = 125.

2 e. TC3 = 500 + 15Q + 1.0Q

MC3 = 15 + 2Q

At Q = 50, MC3 = 15 + 2(50) = 115 2 TC4 = 500 + 5Q + .25Q MC4 = 5 + 0.5Q

At Q = 50, MC4 = 5 + 0.5(50) = 30

With the TC3 cost function, Chillman would be under pressure to raise its price and reduce its output, since MC3 > MR1 at the kink. However, fearing the loss of revenues (if its competitors do not increase their prices), the firm might decide to keep its price at 125.

Similarly, with the TC4 cost function, Chillman would be under pressure to lower its price (and increase output), since MC4 < MR2 at the kink. Again, fearing the loss of revenues if its competitors also reduce their prices, the firm might decide to hold its price at 125. If all firms witnessed similar changes in their cost functions, a new, higher equilibrium price might be established in the first case

(i.e., TC3), and a lower price in the second case (i.e., TC4).

8. a. Player A does not have a dominant strategy. Its optimal decision depends on what player B does. If player B chooses strategy 1, then player A does best by also choosing strategy 1. However, if player B chooses strategy 2, then player A's optimal choice is also strategy 2. b. Player B likewise does not have a dominant strategy. Its optimal decision depends on what player A does. If player A chooses strategy 1, then player B does best by also choosing strategy 1. However, if player A chooses strategy 2, then player B's optimal choice is also strategy 2. Oligopoly 64

9. a. SAMI Strategy Abide Not Abide

Abide $30*; $20 $10; $30 AMC Strategy Not Abide $40; $5 $15; $10

* Payoffs are in millions of dollars. b. The dominant strategy for AMC is to "Not Abide" by the agreement since this strategy yields a guaranteed minimum annual profit of $15 million, regardless of what strategy SAMI selects. c. The dominant strategy for SAMI is to "Not Abide" by the agreement since this strategy yields a guaranteed minimum annual profit of $10 million, regardless of what strategy AMC chooses. d. Each firm should choose the "Abide" strategy since annual profits are higher ($30 million vs. $15 million for AMC and $20 million vs. $10 million for SAMI) than when each firm chooses the "Not Abide" strategy.

Solution to Case Exercise

Cell Phones Displace Mobile Phone Satellite Networks

1. The Managerial Challenge at the start of this Chapter describes the emergence of Nokia as a dominant firm in digital mobile telephony.

2. Motorola might have anticipated better the explosive growth in nonbusiness cell phones and the convergence of mobile telephony, personal digital assistants, cameras, and computer games.