Question

Suppose that the Sacramento River Cats can segment their fans into young fans and senior citizens. Young fans have the demand curve P=120-10G(MR=120-20G). Senior citizens have the demand curve P=60-10G(MR=60-20G). Assume that MC=0. (a)What are the equilibrium price and quantity if the monopoly charges both groups the same price?(b)What are the equilibrium price and quantity for each group if the River Cats can segment the market?(c)Calculate producer surplus in each case.

Answer 1

Solution :-

(a) :-

Young fans have the demand curve :-

P=120-10G(MR=120-20G).

Inverse demand curve for Young fans :-

G = ( 120 - P)/10

Senior citizens have the demand curve :-

P= 60-10G(MR=60-20G).

Inverse demand curve for Senior citizens :-

G = ( 60 - P)/10

Assume that MC = 0

if the monopoly charges both groups the same price Then total demand in the market :-

G' = G + G

= (120 - P)/10 + ( 60 - P)/10

= [( 120 - P) + ( 60 - P) ]/10

= ( 120 - P + 60 - P)/10

G' = ( 180 - 2P)/10

Then,

G' = (90 - P)/5......The inverse demand curve for market

5G' = 90 - P

[ P = 90 - 5G' ]....... demand curve for the market

Now,

Total revenue = P x G'

= ( 90 - 5G') x G'

= 90G - 5G'^2

MR = differentiation of total revenue with respect to G' = 90 - 10G'

We have,

MC = 0

At Optimal condition :-

MR = MC

90 - 10G' = 0

90 = 10G'

G' = 90/10

[G' = 9 ]

equilibrium quantity G' = 9

So, The demand curve equation

P = 90 - 5G'

Put G' = 9

P = 90 - 5 x 9

= 90 - 45

[ P = 45 ]

Equilibrium price P = 45

(b) :-

The equilibrium price and quantity for each group if the River Cats can segment the market :-

When there are different prices for the two segments :-

For Young fans :-

MR =120-20G

MC =0

Optimal condition -

MR = MC

120 - 20G = 0

120 = 20G

G = 120/20

[ G* = 6 ]

So, The equilibrium quantity for Young fans is

G* = 6

Now, The demand curve for Young fans :-

P* = 120 - 10G

Put G = 6

P* = 120 - 10 x 6

= 120 - 60

[ P* = 60 ]

Equilibrium price for Young fans P* = 60.

For senior citizens :-

MR = 60-20G

MC = 0

Optimal condition -

MR = MC

60 - 20G = 0

60 = 20G

G = 60/20

[ G** = 3 ]

Equilibrium quantity for senior citizens G** = 3

Now, The demand curve for senior citizens :-

P** = 60 - 10G

Put G = 3

P** = 60 - 10 x 3

= 60 - 30

[P** = 30 ]

Equilibrium quantity for senior citizens P** = 30

(c) :-

producer surplus in each case.

Here, cost is zero

Then,

Producer surplus = profit

When there was a uniform price in the market :-

Producer surplus = P x G'

We know P = 45, G' = 9

Producer surplus = 45 x 9

= 405

When price is not uniform :-

Producer surplus from Young fans segment

= P* x G*

= 60 x 6........( P* = 60 , G* = 6)

= 360

Producer surplus from senior citizens segment

= P** x G**

= 30 x 3......( P**= 30, G** = 3)

= 90

Total producer surplus = 360 + 90

= 450