Question

Question 1: Demand in an industry is 1200 – Q/10.
There are 20 identical firms in a competitive fringe, each with the cost function:
C(q) = 1000q + q2
.

There is also a dominant firm with the cost function C(q) = b*q, where b is a constant, and b<1000.
(a) What is the maximum possible value of b such that the fringe produces zero output?
(b) Suppose b=500. Work out the equilibrium price and Consumer Surplus.
(c) Suppose b= 1000. Work out the output of the Dominant Firm and of the Competitive Fringe.

Answer 1

Solution :-

(a) :-

Demand in an industry is P = 1200 – Q/10.

C(q) = 1000q + q2..........20 firms, fringe

There is also a dominant firm with the cost function C(q) = b*q

where b is a constant, and b<1000.

Now,

MCi = 2q + 1000

P - 1000 = 2q

Divide both side by 2

qi = P/2 - 500

QF = qi

= 20qi

= 20 ( P/2 - 500)

QF = 10P - 10000

Now, as market demand curve

Q = 12000 - 10P

So, residual demand curve -

QR = 12000 - 10P - 10P + 10000

Q = 22000 - 20P

22000 - Q = 20P

P = ( 22000 - Q)/20

P = 1100 - Q/20

MR = ( 22000 - 2Q)/20

= 1100 - Q/10

So, MC for dominant firm = b

At equilibrium , MR = MC

1100 - Q/10 = b

1100 - b = Q/10

( 1100 - b) x 10 = Q*

Q* = 11000 - 10b

So,

P* = 1100 - Q/20

= 1100 - ( 11000 - 10b)/20

= [ 22000 - ( 11000 - 10b)]/20

P* = (11000 + 10b)/20

Now, individual fringe firm output :-

qi = P*/2 - 500

qi = ( 11000 + 10b)/ 2 x 20 - 500

qi = ( 11000 + 10b)/40 - 500

qi =( 11000 + 10b - 20000)/40 =0

If, 20000 = 11000 + 10b

20000 - 11000 = 10b

9000 = 10b

b = 9000/10

b* = 900

So, the maximum possible value of b = 900.

(b) :-

Suppose b = 500

As b < b* = 900

So, fringe firm do not produce anything

Thus, at equilibrium,

MR = MC = b = 500

1200 - 2Q/10 = 500

1200 - 500 = 2Q/10

700 = 2Q/10

7000 = 2Q

QD = 7000/2

QD = 3500

P = 1200 - Q/10

= 1200 - 3500/10

= 1200 - 350

P = 850

So,

consumer surplus = 1/2 ( 1200 - 850) x 3500

= 1/2 x 350 x 3500

= 175 x 3500

= 612,500

Consumer surplus = $612,500

(c) :-

Suppose b= 1000

b > b*

So, fringe firm do produce positive output level,

P* = (11000 + 10b)/20

= ( 11000 + 10 x 1000)/20.......(b= 1000)

= ( 11000 + 10000)/20

= 21000/20

[ P* = $1050 ]

Q* = 11000 - 10b

= (11000 - 10 x 1000)

= 11000 - 10000

= 1000

[Q* = 1000 ]...... output of dominant firm

Now,

qi = ( 11000 + 10b)/40 - 500

=( 11000 + 10 x 1000)/40 - 500

= ( 11000 + 10000)/40 - 500

=( 21000/40) - 500

= (21000 - 20000)/40

= 1000/40

[ qi = 25 ]

Total fringe firm output supply = 20qi

= 20 x 25

= 500.