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 ]
dA/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.