121A Industrial Organization - Professor Giacomo Bonanno

Ecn 121A: Industrial Organization Professor Giacomo Bonanno

ANSWERS TO PRACTICE PROBLEMS WARNING: If you get stuck trying to do some of the practice problems, it is very likely that the reason is that you don’t know what something means (e.g. "horizontal differentiation"). These practice problems, just like the homework and the exams questions, are about applying concepts and definitions. If you don’t know a definition of course you cannot apply it! Do not peek at the answers to get clues. If you get stuck, you need to identify what it is that you don’t know. Pick up the textbook and/or your notes, review the relevant definitions and concept and then go back to the question. If you still have difficulties, come and see me or the TA. You should look at the answer ONLY IF you are absolutely confident that you answered the question correctly.

1.

(a) Revenue maximized where MR = 0 (MR is Marginal Revenue). Since MR=100−4Q revenue is maximized at Q=25, P=50. (b) (i) MR=100−4Q, MC=4. MR=MC when Q=24, P=52. (ii) Revenue = PQ = (24)(52) = 1248. Cost = 20 + 4Q = 20 + 4(24) = 116. Profits = revenue − cost = 1132. (c) Efficiency requires P=MC, i.e. Q = 48, P = 4. Hence deadweight loss is the area of a right-angled triangle with sides (52−4) = 48 and (48 − 24) = 24. Thus deadweight loss = [48(24)](1/2) = 576.

2. Price

Quantity

MONOPOLIST'S MONOPOLIST'S MONOPOLIST'S REVENUE COSTS PROFITS 16,000 1 (Ann) 16,000 6,000 10,000 14,000 2 (Ann + Bob) 28,000 8,000 20,000 12,000 3(Ann+Bob+Carla) 36,000 10,000 26,000 4 40,000 12,000 10,000 28,000 8,000 5 40,000 14,000 26,000 6,000 6 36,000 16,000 20,000 4,000 7 28,000 18,000 10,000 2,000 8 16,000 20,000 − 4,000 Thus the monopolist will produce 4 computers and sell them for $10,000 each. Now, marginal cost is $2,000. If the firm were forced to produce 8 computers and charge $2,000, the firm would go from a profit of $28,000 to a loss of $4,000. Thus the firm would lose $32,000 in moving from (Q=4,P=10,000) to (Q=8,P=2,000). The first 4 consumers would gain $8,000 each, for a total of $32,000. The new four consumers (consumers 5,6,7,8) would gain a total consumer surplus of $12,000.

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121A Industrial Organization - Professor Giacomo Bonanno

Consumer Ann Bob Carla David

When Q = 4 and P = 10,000 Consumer surplus Total CS : 12,000 Profits: 28,000 16,000 − 10,000 = 6,000 14,000 − 10,000 = 4,000 Social Welfare: 40,000 (SW = CS + Profits) 12,000 − 10,000 = 2,000 10,000 − 10,000 = 0 Total CS = 12,000

Consumer Ann Bob Carla David Emily Frank Gail Henry

When Q = 8 and P = 2,000 Consumer surplus Total CS : 56,000 16,000 − 2,000 = 14,000 Profits: − 4,000 14,000 − 2,000 = 12,000 Social Welfare: 52,000 (SW = CS + Profits) 12,000 − 2,000 = 10,000 10,000 − 2,000 = 8,000 8,000 − 2,000 = 6,000 6,000 − 2,000 = 4,000 4,000 − 2,000 = 2,000 2,000 − 2,000 = 0 Total CS = 56,000

Thus deadweight loss = 52,000 − 40,000 = 12,000 (difference in total surplus or social welfare). Explanation: all the 8 consumers together gain $32,000 + $12,000 = $44,000, while the firm loses 32,000 (goes from a profit of 28,000 to a loss of 4,000). The deadweight loss is equal to consumers' gain minus firm's loss = 44,000 − 32,000 = 12,000. Note that this is precisely the consumer surplus of the new consumers (Emily to Henry).

3.

If the (monthly) demand function is of the form Q = α − P then Rollabybaby's

profit function is π(P) = (P − 35)(α − P). The first-order condition for profit maximization is: π′(P) = α − 2P + 35 = 0. Since the solution is P = 220, we can deduce that α = 405. Thus the demand function is Q = 405 − P. Hence Rollabybaby sells (405 − 220) = 185 pairs of roller skates per month, that is (185)12 = 2,220 per year. Thus Rollabybaby's annual profit is 2,220*(220 − 35) = $410,700. Hence you have good reasons to audit the firm.

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121A Industrial Organization - Professor Giacomo Bonanno

4. Demand function: Q = a - p/b a D( p ) p

600

b

4

c

100

p

a

b

0 .. 2400

p c

2400 2160 1920 1680 1440 1200 960 720 480 240 0

0

60 120 180 240 300 360 420 480 540600 D( p )

Cost function:

C( q ) MC

Inverse demand:

c. q c

given q D( p ) I( q )

find( p )

4. q

2400

(1) PERFECTLY COMPETITIVE INDUSTRY If industry is perfectly competitive, P = MC given I( q ) MC pcq

find( q )

pcp

I( pcq )

575 pcp = 100

Thus in a perfectly competitive industry, price = 100, quantity = 575. Page 3 of 17

121A Industrial Organization - Professor Giacomo Bonanno

pcCS

( 2400

pcp ) . pcq 2 Industry profits are zero (unit cost = marginal cost = price)

Consumer surplus is pcCS = 661250 Total consumer surplus is

pcCS = 661250

(2) NON-DISCRIMINATING MONOPOLY Marginal revenue is:

d ( q. I( q ) ) dq

MR( q )

8. q

MR( q )

2400

Monopoly output given by solution to MR = MC

given MR( q ) MC nmq Thus output is nmq = 287.5 Consumer surplus is:

Monopoly pofits: Total surplus is

575

find( q )

2

and price is

nmp = 1250

( 2400

1250 ) . 287.5

nmCS

nmπ nmTS

2

nmq. ( nmp MC ) nmCS nmπ

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nmp

I( nmq)

nmCS = 165312.5

nmπ = 330625 nmTS = 495937.5

121A Industrial Organization - Professor Giacomo Bonanno

(3) PERFECTLY DISCRIMINATING MONOPOLY Ouput will be where P = MC. Thus dmq

pcq

dmq = 575

Cosnumer surplus is zero. Monopolist's profits are

dmq dmπ

I( q ) dq

( dmq. MC )

dmπ = 661250

0 Can also be computed as the consumer surplus under perfect competition

COMPARISON

Perfect competition

Nondiscriminating monopoly

Perfectly discriminating monopoly

Price

Output

pcp = 100

pcq = 575

nmp = 1250

CS

Profits

pcCS = 661250

0

pcCS = 661250

nmπ = 330625

nmq = 287.5

nmCS = 165312.5

N/A

Total surplus

dmq = 575

5.

0

nmTS = 495937.5

dmπ = 661250 dmπ = 661250

(a) A two-part tariff involves a price equal to marginal cost. In this case P = 10. At this price a type 1 consumer demands 45 units while a type 2 consumer demands 70 units. For a type 1 consumer, the total willingness to pay for 45 units is given by the area under the demand function between 0 and 45. Thus it is equal to (see shaded area in figure below) (100 − 10)45 + 45(10) = 2,475 2 By paying $10 for each unit, the consumer spends $450, hence enjoys a surplus of 2,475 − 450 = 2,025. Thus the fixed fee can be set equal to this amount to extract all the consumer surplus. Thus for type one consumer: fixed fee = 2,025, price per unit = 10.

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121A Industrial Organization - Professor Giacomo Bonanno

P 100

10 Q 0

45

50

Similarly, for a type 2 consumer the total willingness to pay for 70 units is given by the area under the demand curve between 0 and 70. Thus it is equal to (see figure below) (80 − 10)70 + 70(10) = 3,150 2 By paying $10 for each unit, the consumer spends $700, hence enjoys a surplus of 3,150 − 700 = 2,450. Thus the fixed fee can be set equal to this amount to extract all the consumer surplus. Thus for type one consumer: fixed fee = 2,450, price per unit = 10.

(b) First we have to check, for each type of consumer which package the consumer would choose. willingness to pay for 50 units (package 1)

Type 1

area under demand curve from 0 to 50

cost of package 1

2,500

surplus from package 1

0

willingness to pay for 40 units (package 2) area under demand curve from 0 to 40

= 2,400

100(50) = 2500 2

Type 2

area under demand curve from 0 to 50 (note when Q = 50, P = 30): ( 80 − 30 ) 50 2

2,500

250

area under demand curve from 0 to 40

= 2,400

cost of package 2

surplus from package 2

2,200

200

2,200

200

+ 30(50)

= 2,750

Thus type 1 consumers choose package 2 and type 2 consumers choose package 1.

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121A Industrial Organization - Professor Giacomo Bonanno

The profit (gross of fixed cost) from a type 1 consumer (i.e. from a package 2) is: 2,200 − 10(40) = 1,800. The profit (gross of fixed cost) from a type 2 consumer (i.e. from a package 1) is: 2,500 − 10(50) = 2,000. Thus total profits are: 100 (1,800) + 50 (2,000) − 200 (fixed cost) = $279,800.

6. Demand from group I consumers:

DI( p )

Demand for group II consumers:

p

p p

20

DII( p )

24

p 2 p 2

DI( p )

DII( p )

0 .. 48 48 44 40 36 32 28 24 20 16 12 8 4 0

(1)

0

4

8

12

16

20

24

DI( p ) , DII( p )

CASE 1 Let DA denote aggregate demand. Then

P  D II = 24 − 2  DA(P) =  D + D = 44 − P II  I  DA(40) = 4. Thus the inverse aggregate demand function is

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if P > 40 if P ≤ 40

44

p

121A Industrial Organization - Professor Giacomo Bonanno

48 − 2Q Q=  44 − Q

if Q < 4 if Q ≥ 4

(2) Thus Marginal Revenue is given by 48 − 4Q MR =  44 − 2Q

if Q < 4 if Q ≥ 4

48 48 44 40 36 Inv( q )

32 28

MR( q ) 24 4

20 16 12 8 4 0 0

0 0

4

8

12

16

20

24

28

32

36

40

q

44 44

The dashed line is DA, the continuous line is MR and the thick horizontal line is Marginal Cost = 4. It is clear from the above picture that MR = MC at a level of output greater than 4. Thus to find profit-maximizing level of output, solve for Q the following 44 − 2 Q = 4

(3)

The solution is: Q = 20 and the corresponding price is P = 24.

(4)

Group I consumers buy: DI(24) = 8 units. Their consumer surplus is: (40 − 24) 8 = 64 2

Group II consumers buy: DII(24) = 12 units. Their consumer surplus is: (48 − 24) 12 = 144 2

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121A Industrial Organization - Professor Giacomo Bonanno

The monopolist’s profits are:

π = (24 − 4) 20 = 400.

Thus total surplus is: 64 + 144 + 400 = 608.

CASE 2 (5) Inverse demand for group I is: PI =40 − 2 QI. Inverse demand for group II is: PII = 48 − 2 QII. Thus MRI = 40 − 4 QI and MRII = 48 − 4 QII. Hence the monopolist will choose QI to solve MRI = MC and choose QII to solve MRII = MC (with MC = 4). This gives QI = 9 and QII = 11. The prices will be PI = 22 and PII = 26. (6)

Consumer surplus for group I is:

Consumer surplus for group II is:

(40 − 22) 9 = 81 2

(48 − 26) 11 = 121 2

The monopolist’s profit is: 9 (22 − 4) + 11 (26 − 4) = 404. Total surplus is: 81 + 121 + 404 = 606.

(7) Group I consumers and the monopolist gain from price discrimination, Group II consumers and society lose.

7.

(a) If TOYOTA and Michelin are integrated into a single firm, the cost of producing one car is (3+6)=9. The firm would be a monopolist with profit function π(p) = (p − 9) (40 −4p) Monopoly price is given by the solution to dp = 40 − 4p + (p − 9)(−4) = 0. dp Thus: p = 9.5 q = 2.

(b) If Michelin and TOYOTA are separate firms, let w be the price quoted by Michelin to TOYOTA. Then TOYOTA's unit cost of production is w+6. TOYOTA is a monopolist in the car market and will choose p to maximize: π(p) = [p − (w + 6)] (40 − 4p). Solving

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121A Industrial Organization - Professor Giacomo Bonanno

dπ = 40 − 4p + (p − w − 6)(−4) = 0 dp we obtain p=8+

w 2

q = 8 − 2w. Thus if Michelin sets a price of w, TOYOTA will buy (8 − 2w) units from Michelin at that price. Hence Michelin will choose w to maximize (recall that Michelin's unit cost of production is 3): π(w) = (w − 3) (8 − 2w). Solving dπ = 8 − 2w + (w − 3)(−2) = 0 dw we obtain w = 3.5 with corresponding quantity and price of q = 8 − 2(3.5) = 1 p=8+

3.5 = 9.75. 2

Thus consumers are better off if Michelin and TOYOTA are integrated (they pay a lower price and buy more cars).

8.

(a) If IBM and NEC are integrated into a single firm, the cost of producing one

complete PC is (2+4)=6. The firm would be a monopolist with profit function π(p) = (p − 6) (200 − p) Monopoly price is given by the solution to ∂π = 200 − p − p + 6 = 0. ∂p Thus: p = 103

and

q = 97.

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121A Industrial Organization - Professor Giacomo Bonanno

(b) If NEC and IBM are separate firms, let w be the price quoted by NEC to IBM. Then IBM's unit cost of production is w+4. IBM is a monopolist in the computer market and will choose p to maximize: π(p) = [p − (w + 4)] (200 − p). dπ = 200 − p − p + w + 4 = 0 dp w w p = 102 + and q = 98 − . 2 2

Solving we obtain

w  Thus if NEC sets a price of w, IBM will buy 98 -  units from NEC at that price. 2  Hence NEC will choose w to maximize (recall that NEC's unit cost of production is 2): w  π(w) = (w − 2) 98 - . 2  dπ w w   = 98 -  − +1=0 dw  2 2 w = 99

Solving we obtain

with corresponding quantity and price of q = 48.5 and p = 200 - 48.5 = 151.5. Thus consumers are better off if NEC and IBM are integrated (they pay a lower price and buy more computers).

9.

Costs are given by:

FIRM 1

FIRM 2

q

TC

q

TC

1

5

1

6

2

9

2

8

3

13

3

10

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121A Industrial Organization - Professor Giacomo Bonanno

(a) Firm 2' output 1

2

3

Firm

1

15, 14

13, 28

5, 20

1's

2

27, 12

11, 12

7, 14

output

3

17, 4

11, 8

−1, 2

(b) Firm 1 does not have a dominant strategy (for example, if firm 2 chooses 1 then it is best for firm 1 to choose 2, while if firm 2 chooses 2 then it is best for firm 1 to choose 1). Firm 2 also does not have a dominant strategy (best reply to 1 is 2, while best reply to 2 is 3).

(c-d) The Nash equilibria are (1,2) and (2,3), with corresponding profits (13,28) and (7,14). Both firms prefer the first, thus it is likely that they would coordinate on (1,2).

10. output of output of firm 1 firm 2 5 5 5 10 10 5 10 10

total output 10 15 15 20

price

firm 1's revenue 450 425 850 800

90 85 85 80

firm 2's revenue 450 850 425 800

firm 1's costs 30 30 55 55

firm 2's costs 30 55 30 55

firm 1's profits 420 395 795 745

Thus we have the following bimatrix game: Firm 2’s output 5 10 Firm 1’s output

5

420 , 420

395 , 795

10

795 , 395

745 , 745

Producing 10 units is a dominant strategy for both firms. (10,10) is the unique Nash equilibrium.

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firm 2's profits 420 795 395 745

121A Industrial Organization - Professor Giacomo Bonanno

11.

(a-b) The profit function of firm 2 is: π2(q1,q2) = q2[160 − (q1+q2)]. ∂π2 = 160 − q1− 2q2 . Setting it equal to zero and solving for q2 we get firm 2's ∂q2 160 - q1 reaction function: R2(q1) = . Thus if firm 1 announces an output of 10, 2 firm 2 will choose output R2(10) = 75.

(c) The profit function of firm 1 is: π1(q1,q2) = q1[160 − (q1+q2)]. Substituting R2(q1) for q2 we obtain 160 - q1     π1(q1) = q1 160 - q1+ 2    Thus π′(q1) = 160 − 2q1 − 80 + q1. Setting it equal to zero and solving for q1 gives q1 = 80.

12.

(a) The profit function of firm 1 is given by π1(q1,q2) = q1 [2000 − 2(q1+q2)] − 560 q1 − 80,000

The reaction function of firm 1 is obtained by solving the equation q1 = 360 −

∂π1 = 0, which gives ∂q1

q2 . 2

Similarly, the reaction function of firm 2 is given by q2 = 360 −

720

output of firm 2

reaction curve of firm 1

360 240

reaction curve of firm 2 0

240

360

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output of firm 1 720

q1 . 2

121A Industrial Organization - Professor Giacomo Bonanno

(b) The Cournot-Nash equilibrium is given by: q1 = q2 = 240 Q = 480 P = 1040 π1 = π2 = 115,200 − 80,000 = 35,200.

(c) A monopolist would set Q = 360 and P = 1280. Her profits would be π = 259,200 − 80,000 = 179,200. The demand curve is given by:

P CS under monopoly

2000

1280

CS under duopoly

1040 Q 0

360

480

1000

Thus consumer surplus under monopoly would be (2000 − 1280)(360)/2 = 129,600, while consumer surplus under duopoly is (2000 − 1040)(480)/2 = 230,400. Hence social welfare under monopoly is 129,600 + 179,200 = 308,800, and social welfare under duopoly is 230,400 + 2(35,200) = 300,800. Thus social welfare is higher under monopoly than under duopoly. The reason is that under duopoly the fixed cost has to be paid twice and in this case the fixed cost is larger than the gain in consumer surplus obtained by switching from monopoly to duopoly.

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121A Industrial Organization - Professor Giacomo Bonanno

13.

Inverse demand is P = 300 − 3Q. (a) π1(q1,q2) = q1[300 − 3(q1+q2)] − 2q1 − 150. π2(q1,q2) = q2[300 − 3(q1+q2)] − 2q2 − 150

(b)

(c)

The Cournot equilibrium is given by the solution to the system of equations

(d) and

∂π1 = 300 − 6q1− 3q2 − 2. Setting it equal to zero and solving for q1 ∂q1 298 - 3q2 gives: R1(q2) = . 6 238 R1(20) = = 39.67. 6 ∂π1 =0 ∂q1

∂π2 = 0. The solution is: q1 = q2 = 33.11, P = 101.33, π1 = π2 = 3139. ∂q2

14.

3 2 p > 0 (provided p < ) and D2 = 0. 2 3 Thus all consumers would go to firm 1. Therefore the products are vertically differentiated (product 1 is considered by consumers to be of higher quality than product 2).

15.

This is a case of horizontal differentiation: if prices are equal, some consumers will prefer the faster computer, while others will prefer the computer with the larger fixed disk.

16.

1 (a) Let firm 1 be located at 4 and firm 2 at 1. Let p1 be the price of firm 1 and p2 the price of firm 2. Let x be the consumer who is indifferent between buying from firm 1 and buying from firm 2. Then x must satisfy:

Setting p1 = p2 = p we obtain: D1 = 1 −

1

p1 + 2 (x − 4 ) = p2 + 2 (1 − x). Hence x =

p2 - p1 5 + 8 . Demand for firm 1 is x, while demand for firm 2 is 4

(1−x). Thus: D1(p1,p2) =

p2 - p1 5 + 8 4

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121A Industrial Organization - Professor Giacomo Bonanno

p1 - p2 3 D2(p1,p2) = 8 + . 4

(b) The profit (= revenue) function of firm 1 is given by π1=p1D1 and the profit (= revenue) function of firm 2 is given by π2=p2D2. The Nash equilibrium is obtained ∂π1 ∂π2 by solving the system of equations = 0 and = 0. ∂p1 ∂p2 ∂π1 p2 - p1 p1 5 = + 8 − =0 4 4 ∂p1 ∂π2 p1 - p2 p2 3 = 8 + − = 0. 4 4 ∂p2 13

11

The solution is p1 = 6 and p2 = 6 . Thus the two gas stations would charge different prices.

17.

If the rate of interest is r = 0.35, the discount rate is δ =

1 = 0.74. If firm 1 1+r

produces 1 unit always, then its profit is 12 every period and the present value of the profit stream is: 12 + δ 12 + δ2 12 + δ3 12 + ... = 12 ( 1 + δ + δ2 + δ3 + ... ) = 12

1 = 46.285. 1−δ

If firm 1 produces 1 unit for the first 9 periods then 2 at date 10, then it had better produce 2 for every date t > 10 (because this is what firm 2 will do). Thus its profit stream will be: 12 for periods 1-9, 50 in period 10, 2 ever after. The present value of this profit stream is: 12 + δ 12 + δ2 12 + δ3 12 + ... + δ8 12 + δ9 50 + δ10 2 + δ 11 2 + δ 12 2 + ... = = 12 (1 + δ + δ2 + δ3 + ... + δ8) + δ9 50 + 2 δ10 ( 1 + δ + δ2 + δ3 + ... ) =  1 − δ9  1  + δ9 50 + 2 δ10 = 12  = 46.918. 1 − δ 1 − δ   Thus firm 1 is better off switching to 2 units from date 10 onwards.

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121A Industrial Organization - Professor Giacomo Bonanno

18.

CASE 1: the firm is protected from entry (e.g. by a patent)

If the firm does not invest in R&D, its profit function is: π(q) = q (40 − 2 q) − ( 8q + 45). ∂π = 0 gives: q = 8, P = 24, π = 83. Solving ∂q If the firm invests in R&D, its profit function will be: π(q) = q (40 − 2 q) − ( 4q + 45) ∂π = 0 gives: q = 9, P = 22, π = 117 (this is without taking into account the ∂q investment cost of $36). Thus lowering costs leads to an increase in profits of 117 − 83 = $34; it is not worth, since it requires spending $36 (investment in R&D involves a loss of $2). Alternatively, we could have compared the profit net of investment cost, namely 117 − 36 = 81, to 83 (investment in R&D involves a loss of $2). Solving

CASE 2: the firm is not protected from entry If the incumbent does not invest in R&D and there is entry, then the incumbent’s profit function will be πi = qi(40 − 2qi − 2qe) − ( 8qi + 45) and the entrant’s profit function will be πe = qe(40 − 2qi − 2qe) − ( 8qe + 45). The Cournot equilibrium is given by the ∂π i ∂πe solution to = 0 and = 0. The solution is qi = qe = 16/3 with πi = πe = 11.89. ∂qi ∂qi Since πε > 0, the potential entrant would decide to enter. If the incumbent invests in R&D and there is entry, then the incumbent’s profit function will be πi = qi(40 − 2qi − 2qe) − ( 4qi + 45) and the entrant’s profit function will be πe = qe(40 − 2qi − 2qe) − ( 8qe + 45). The Cournot equilibrium is given by the solution to ∂π i ∂πe = 0 and = 0. The solution is qi = 20/3 and qe = 14/3 with πi = 43.89 (gross of ∂qi ∂qi the R&D cost, thus net πi = 7.89) and πe = − 1.44. Since πε < 0, the potential entrant would decide not to enter. Thus, anticipating all this, the incumbent has to choose between not investing in R&D (thus entry, thus a profit of 11.89) and investing in R&D (thus no entry, thus profit of 117 − 36 = 81). Clearly the incumbent will choose investment in R&D.

(b) This is a strategic choice, because if there were no threat of entry the incumbent would not find it profitable to invest in R&D (as shown above). Hence the investment in R&D is motivated entirely by the desire to deter entry, i.e. to limit competition in this industry. Hence, at least in principle, the firm could be found in violation of antitrust laws.

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