Question

A monopolist with constant marginal cost 10 per unit supplies 2 markets. The inverse demand equation in market 1 is p1 = 100 - Q1. The inverse demand equation in market 2 is

p2 = 50 - 50/N2 * Q2; where N2 is the number of individuals in market.

(a) Consider first the case that the monopolist can set different prices for the two markets. Find monopoly price, output, and consumer surplus in market 1. Find monopoly price, output, and consumer surplus in market 2 if N2 = 50. Find monopoly price, output, and consumer surplus in market 2 if N2 = 25.

(b) Consider next the case that the monopolist must set the same price for both markets. Find monopoly price, output, sales in both markets, and consumer surplus if N2 = 50. Find monopoly price, output, sales in both markets, and consumer surplus if N2 = 25.

(c) If the firm is allowed to price discriminate, how does that affect consumer surplus in each market if N2 = 50? If N2 = 25?

Answer 1

a.i. Given : Marginal cost MC = 10

The inverse demand curve in market 1 is p₁ = 100-Q₁

The vertical and horizontal intercepts of the demand curve in market 1 are 100 each.

Now, total revenue TR₁ = p₁*Q₁ = 100Q₁-Q₁²

Then, marginal revenue MR₁ = dTR₁/dQ₁ = 100-2Q₁

Now, for equilibrium in market 1, MR₁ = MC

or, 100-2Q₁ = 10

or, 2Q₁ = 90

or, Q₁ = 45 units is the equilibrium output

and p₁ = 100-45 = $55 per unit is the equilibrium price.

and Consumer surplus = 1/2*(vertical intercept of the demand curve - equilibrium price)*equilibrium output

or, Consumer surplus = 1/2*(100-55)*45

or, Consumer surplus = 1/2*45*45

or, Consumer surplus = $1,012.5

ii. Similarly, the inverse demand curve in market 2 is p₂ = 50-(50/N₂)*Q₂

When N₂ = 50, p₂ = 50-(50/50)*Q₂ or, p₂ = 50-Q₂

Then, the vertical and horizontal intercepts of the demand curve in market 2 are 50 each.

Now, total revenue TR₂= p₂*Q₂ = 50Q₂-Q₂²

Then, marginal revenue MR₂ = dTR₂/dQ₂ = 50-2Q₂

Now, for equilibrium, MR₂ = MC

or, 50-2Q₂ = 10

or, 2Q₂ = 40

or, Q₂ = 20 units is the equilibrium output in market 2 when N₂ = 50

and p₂ = 50-20 = $30 per unit is the equilibrium price in market 2 when N₂ = 50

and consumer surplus = 1/2*(vertical intercept of the demand curve - equilibrium price)*equilibrium output

or, Consumer surplus = 1/2*(50-30)*20

or, Consumer surplus = 1/2*20*20

or, Consumer surplus = $200

iii. Again, when N₂ = 25, p₂ = 50-(50/25)*Q₂ or, p₂ = 50-2Q₂

Then, the vertical intercept and the horizontal intercept of the demand curve in market 2 are 50 and 25 respectively.

Now, total revenue TR₂= p₂*Q₂ = 50Q₂-2Q₂²

Then, marginal revenue MR₂ = dTR₂/dQ₂ = 50-4Q₂

Now, for equilibrium, MR₂ = MC

or, 50-4Q₂ = 10

or, 4Q₂ = 40

or, Q₂ = 10 units is the equilibrium output in market 2 when N₂ = 25

and p₂ = 50-(2*10) = $30 per unit is the equilibrium price in market 2 when N₂ = 25

and consumer surplus = 1/2*(vertical intercept of the demand curve - equilibrium price)*equilibrium output

or, Consumer surplus = 1/2*(50-30)*10

or, Consumer surplus = 1/2*20*10

or, Consumer surplus = $100

Thus, if the monopoly charges different prices, then it will charge $45 in market 1 and $30 in market 2.

b.i. If monopoly charges same price (say p) in both markets,

Then, aggregate demand Q = Q₁+Q₂

or, Q = 100-p+50-p (where p=p₁=p₂ and N₂ = 50)

or, Q = 150-2p

or, p = 75-(Q/2)

Here, the vertical intercept of the aggregate demand curve is 75.

Now, total revenue TR = p*Q = 75Q-(Q²/2)

Then, marginal revenue MR = dTR/dQ = 75-Q

Now, for equilibrium, MR=MC

or, 75-Q=10

or, Q = 65 units is the equilibrium output at same price in both markets when N₂ = 50

and p = 75-(65/2) = $42.5 is the equilibrium price when N₂ = 50

and consumer surplus = 1/2*(vertical intercept of the aggregate demand curve - equilibrium price)*equilibrium output

or, consumer surplus = 1/2*(75-42.5)*65

or, consumer surplus = $1,056.25

ii. Similarly, when N₂ = 25,

aggregate demand Q = Q₁+Q₂

or, Q = 100-p+25-(p/2) (where p=p₁=p₂ and N₂ = 25)

or, Q = 125-(3p/2)

or, p = 2/3*(125-Q)

Here, the vertical intercept of the aggregate demand curve is 83.33.

Now, total revenue TR = p*Q = 2/3*(125Q-Q²)

Then, marginal revenue MR = dTR/dQ = 2/3*(125-2Q)

Now, for equilibrium, MR=MC

or, 2/3*(125-2Q)=10

or, 125-2Q = 15

or, 2Q = 110

or, Q = 55 units is the equilibrium output at same price in both markets when N₂ = 25

and p = 2/3*(125-55) = $46.67 is the equilibrium price when N₂ = 25

and consumer surplus = 1/2*(vertical intercept of the aggregate demand curve - equilibrium price)*equilibrium output

or, consumer surplus = 1/2*(83.33-46.67)*55

or, consumer surplus = $1,008.15

c. If the monopolist price discriminates,

when N₂ = 50, total consumer surplus in market 1 and market 2 is $1,012.50+$200 = $1,212.50

and when N₂ = 25, total consumer surplus in market 1 and market 2 is $1,012.50+$100 = $1,112.50.

Thus, consumer surplus falls.