Question

A company has monopoly rights to sell the popular video “Boyfights”. Suppose there are two groups of buyers of Boyfights. Demand for the respective groups is given by the following equations:

Group I: P1 = 10 – Q1 Group II: P2 = 8 – 2Q2

Where Q1 and Q2 are the number of Boyfights movies sold to each group. The total cost function for making each copy of the movie is TC = Q, where Q = Q1 + Q2. The marginal cost function is thus MC = 1.

Now suppose that the company can identify buyers from each group. Also assume that buyers from one group cannot resell the good to buyers from the other group. The monopolist will practice third-degree price discrimination by setting separate prices, P1 and P2 for Group I and Group II respectively.

a) Calculate the optimal P₁ and P₂

b) What is the total deadweight loss under the third-degree price discrimination (relative to perfect competition)?

Answer 1

a) From the given information, it is clear that the demand function for Group I is P1 = 10 – Q1 and that for Group II is P2 = 8 – 2Q2. The marginal cost (MC) is identical for both the groups because there is only 1 producers. Since Total cost (TC) = Q, MC = dTC/dQ = 1.

Total Revenue of group I (TR1) = P1*Q1 = 10Q1 – Q1²

Similarly that of group II (TR2) = P2*Q2 = 8Q2 – 2Q2²

Therefore Marginal Revenue of group I (MR1) = dTR1/dQ1 = 10-2Q1

Similarly, MR2 = dTR2/dQ2 = 8-4Q2

Since 3^rd degree price discrimination involves charging different prices to different groups, each group’s marginal revenue is equal to the marginal cost.

Thus, for group 1,

10-2Q1 = 1

Or 2Q1 = 9 or Q1 = 4.5 units. Therefore, P1= 10-4.5 or P1 = 5.5.

For group 2,

8-4Q2 = 1

Or 4Q2 = 7 or Q2 = 1.75 units. Therefore, P1= 8-3.5 or P2 = 4.5.

Thus, optimal P1= 5.5 and optimal P2 = 4.5.

b) We know that under perfect competition, profit maximization requires P=MC. Under perfect competition (PC), there is no deadweight loss as the entire surplus is divided between consumers’ surplus (CS) (below the demand curve and above the price) and producer’s surplus (PS) (above the supply curve and below the price).

The output that would’ve been produced in group I under PC is:

P1 = MC Or 10 – Q1 =1 or Q1 = 9units. This is shown as Q1* in figure 1.

Similarly, the output that would’ve been produced in group II under PC is:

P2 = MC or 8 – 2Q2= 1 Or 2Q2 = 7 or Q2 = 3.5. This is shown as Q2* in figure 2.

The deadweight loss (DWL) in both the markets is represented by the area of the triangle shaded in orange.

Formula wise, it is:

½ * (base) * (height) where base = output under PC minus the output under 3^rd degree price discrimination and height = Price under third degree price discrimination minus the price under PC (which is equal to marginal cost).

Thus, for group I, DWL1 = ½ * (base) * (height)

Or DWL1 = ½ * (9-4.5) * (5.5-1)

Or DWL1 = 10.125.

This is labelled in Fig 1 as triangle ABC.

Similarly, for group I, DWL2 = ½ * (base) * (height)

Or DWL1 = ½ * (4.5-1) * (3.5-1.75)

Or DWL1 = 3.0625.

This is labelled in Fig 2 as triangle A’B’C’.

Therefore total DWL = DWL1 + DWL2 = 10.125.+ 3.0625 = 13.1875

Therefore total deadweight loss in the monopoly market under third degree price discrimination is 13.1875 as opposed to ZERO deadweight loss under perfect competition.