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In: Economics

Consider a differentiated Bertrand market with three firms, whose demand curves are: Q1 = 300-10P1+3P2 +2P3...

Consider a differentiated Bertrand market with three firms, whose demand curves are:

Q₁ = 300-10P₁+3P₂ +2P₃

Q₂ = 300-8P₂ +2P₁ + P₃

Q₃= 50-3P₃+P₁+P_2.

Firms 1 and 2 both have marginal costs of 10 and Firm 3 has a marginal cost of 15.

P1 = 26.80; P2 = 28.66; P3 = 25.07.

a. Calculate the market shares of each product

If Firm 2 and 3 merge Firm 3’s marginal cost will fall to 10 as a result of the merger.

b. Calculate the UPPI for the merging firms

c. Calculate post-merger prices and profits for all three firms.

d. How are firms and consumers affected by the merger.

Expert Solution

*Answer:

1.

Q1=300-10P1+3P2+2P3

Q2=300-8P2+2P1+P3

Q3= 50-3P3+P1+P2

MC1=10

MC2=10

MC3=15

a)

Profit of firm 1= Pr1= P1 x Q1 - Q1 x MC1

Pr1= P1(300-10P1+3P2+2P3)-(300-10P1+3P2+2P3)10

Pr1= 300P1-10P12+3P1P2+2P1P3-3000+100P1-30P2-20P3

Differentiate with respect to P1

dPr1/dP1= 300-20P1+3P2+2P3+100=0

(400+3P2+2P3)/20= P1 reaction curve of firm 1(BRS 1)

Profit of firm 2= Pr2= P2 x Q2 - Q2 x MC2

Pr2= P2(300-8P2+2P1+P3)-(300-8P2+2P1+P3)10

Pr2= 300P2-8P22+2P1P2+P3P2-3000+80P2-20P1-10P3

Differentiate with respect to P2

dPr2/dP2= 300-16P2+2P1+P3+80=0

P2= (380+2P1+P3)/16 reaction curve of firm 2(BRS 2)

Profit of firm 3= Pr3= P3 x Q3 - Q3 x MC3

Pr3= P3( 50-3P3+P1+P2)-( 50-3P3+P1+P2)15

Pr3= 50P3-3P32+P1P3+P2P3-750+45P3-15P1-15P2

Differentiate with respect to P3

dPr3/dP3= 50-6P3+P1+P2+45=0

P3= (95+P1+P2)/6 reaction curve of firm 3 (BRS 3)

Solve BRS 3 by putting value of P2 from BRS of firm 2

P3= (95+P1+P2)/6

6P3= [95+P1+(380+2P1+P3)/16)]

16 X 6P3= 1520+16P1+380+2P1+P3

95P3= 1900+18P1

P3*= (1900+18P1)/95

Put P3* into BRS of firm 2

P2=(380+2P1+P3)/16

16P2=380+2P1+(1900+18P1)/95

95x16P2= 38000+208P1

P2*= (38000+208P1)/1520

Use P1* and P2* in BRS 1

P1= (400+3P2+2P3)/20

20P1= 400+3[(38000+208P1)/1520]+2[(1900+18P1)/95 ]

144400x20P1= 57760000+10830000+59280P1+5776000+54720P1

144400x20P1= 74366000+114000P1

2774000P1= 74366000

P1*= 26.80

Use this in P2* and P3*

P2*= (38000+208P1)/1520= (38000+208x26.80)/1520= 28.66

P3*= (1900+18P1)/95 =(1900+18x26.80)/95 = 25.07

Use P1*, P2* and P3* in quantities:

Q1=300-10P1+3P2+2P3= 168

Q2=300-8P2+2P1+P3= 149

Q3= 50-3P3+P1+P2= 30

b) Market share= Quantity/Total market quantity

Total market quantity= 168+149+30= 347

Market share of firm 1= 168/347= 0.48

Market share of firm 2= 149/347= 0.43

Market share of firm 3= 0.090

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Comment

Rahul Sunny answered 2 years ago

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