Question

A perfectly competative market with demand Qd=80-P. Let there be 4 firms (indexed by i=1,...,4), each with a cost curve C(qi)=6qi+3/2qi^2.

A) The firms interact in a competative market. What is the equilibrium p and q?

B) Firms 1 and 2 merge and become a firm with a reduced cost function C(qi)=1/2qi^2 while remaining fringe firms have the same cost functions as earlier. The market is still perfectly competative. What is the merged firms output, and there is no price ceiling.

C) Do consumer prefer the market before or after the merger, why?

Please show step by step

Answer 1

a)

C(qi)=6qi+3/2qi^2

Marginal cost each firm=MCi=dC(qi)/dqi=6+3qi

In perfect competition, firms sets output level such that MC=Market price

So,

P=6+3qi

or

3qi=-6+P

qi=-2+(1/3)P

This is the supply curve of each individual firm, There are 4 firms in industry.Total supply is given by

Qs=4*(-2+(1/3)P)=-8+(4/3)P

Put Qd=Qs for equilibrium

80-P=-8+(4/3)P

88=(7/3)P

P=(88*3/7)=37.71429

Qd=80-P=80-37.71429=42.28571

Qs=-8+(4/3)P=-8+(4/3)*37.71429=42.28572

Equilibrium price=37.71429
Equilibrium quantity=42.28572

Output of each firm=42.28572/4=10.57143

B)

C(qi)=1/2qi^2

Marginal Cost of merged firm=MC_M=dC(qi)/dqi=qi

Since every firm is a price taker. MC_M=P

P=qi

We have derived supply function of other firms in part (a)

qi=-2+(1/3)P

Market Supply=P+2*(-2+(1/3)P)=P-4+(2/3)P=-4+(5/3)P

Set Qd=Qs

80-P=-4+(5/3)P

84=(8/3)P

P=84*3/8=31.50

Qd=80-P=80-31.50=48.50

Qs=-4+(5/3)P=-4+(5/3)*31.50=48.50

Merged firm supply function is

P=qi

So, merged firm output is qi=P=31.50

C)

Market price has decreased and equilibrium quantity has increased. There is an increase in consumer surplus. So, consumers will prefer merger.