Question

In: Economics

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

1. 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+P2.

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

a. Calculate the equilibrium prices and quantities.

b. Calculate the market shares.

c. By specific reference to the HMGLs, would the two lead federal agencies be likely to regard a merger between Firm 2 and Firm 3 as one that would likely raise serious competitive concerns?

d. Calculate the UPPIs for the merging parties. Assume that Firm 3’s marginal cost falls to 10 as a result of the merger.

Expert Solution

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= 0.09

Rahul Sunny answered 2 years ago

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