Question

A monopolist is deciding on the quantity of output to produce in two different countries. Demand for the two countries are: Q1=12-P1  Q2=12-2P2 Q2 = 12 − 2P2 ATC = MC = $4 a. (10) What are price, output, and profits, if the monopolist can price discriminate b. (10) What are price, output, and profits,if the law prohibits charging different prices in the two countries? c. (5) Suppose that the monopolist could adopt a two-part tariff, what pricing policy should the firm follow? How do the profits of the monopolist compare to the profits arrived at in part (a) and part (b) above? Under what circumstances is the firm likely to use the two-part tariff pricing?

Answer 1

We have the following information

Demand equation of country 1: Q₁ = 12 – P₁ or P₁ = 12 – Q₁

Demand equation of country 2: Q₂ = 12 – 2P₂ or P₂ = 6 – 0.5Q₂

Marginal cost is given as: MC = $4

Part a) Monopolist carries out price discrimination. In this situation the equilibrium for the monopolist will be the point where the marginal revenue from two countries (MR₁ and MR₂) is equal to the marginal cost

MR₁ = MR₂ = MC

Total Revenue from Country 1 = Price × Quantity

TR₁ = P₁ × Q₁

TR₁ = (12 – Q₁)Q₁

TR₁ = 12Q₁ – Q²₁

MR₁ = ΔTR₁/ΔQ₁ = 12 – 2Q₁

Total Revenue from Country 2: TR₂ = P₂ × Q₂

TR₂ = (6 – 0.5Q₂)Q₂

TR₂ = 6Q₂ – 0.5Q²₂

MR₂ = ΔTR₂/ΔQ₂ = 6 – Q₂

MR₁ = MC

12 – 2Q₁ = 4

Equilibrium quantity in the case of Country 1: Q₁ = 4

MR₂ = MC

6 – Q₂ = 4

Equilibrium quantity in the case of Country 2: Q₂ = 2

P₁ = 12 – Q₁

P₁ = 12 – 4

Equilibrium price in the case of Country 1: P₁ = 8

P₂ = 6 – 0.5Q₂

P₂ = 6 – (0.5 × 2)

Equilibrium price in the case of Country 2: P₂ = 7

Total Profit = Total Revenue₁ + Total Revenue₂ – Total Cost

Total Revenue₁ = 8 × 4 = 32

Total Revenue₂ = 7 × 2 = 14

Total Cost = 4 × (4 + 2) = 4 × 6 = 24

Total Profit = 32 + 14 – 24

Total Profit = 22

Part b) Now, it is given that the Monopolist cannot discriminate. This means that the Monopolist faces a single demand curve.

The single demand curve can be derived by horizontal summation of the two demand curve equations.

P₁ + P₂

12 – Q₁ + 6 – 0.5Q₂

P = 18 – 1.5Q the single demand curve being faced by the Monopolist

P = Price

Q = Total Quantity (Q₁ + Q₂)

Total Revenue (TR) = Price × Quantity

TR = (18 – 1.5Q)Q

TR = 18Q – 1.5Q²

MR = ΔTR/ΔQ = 18 – 3Q

Equating the MR with the MC we get

18 – 3Q = 4

Equilibrium output = 4.67

P = 18 – 1.5Q

P = 18 – (1.5 × 4.67)

Equilibrium price = 11

Π = Total Revenue – Total Cost

Total Revenue = 4.67 × 11 = 51.37

Total Cost = 4 × 11 = 44

Π = 51.37 – 44

Π = 7.37

Part c) Now it is given that the Monopolist adopts two-part tariff system

Under the two-part tariff system, the Monopolist will set the price equal to the marginal cost and add a fixed fee for each user that will entitle the person to make further purchases.

Now we have two countries with the following demand equations

Demand equation of country 1: Q₁ = 12 – P₁ or P₁ = 12 – Q₁

Demand equation of country 2: Q₂ = 12 – 2P₂ or P₂ = 6 – 0.5Q₂

Marginal cost is given as: MC = $4

We will first decide on the fixed fee which cannot be larger than the consumer surplus of the customer with the smallest consumer surplus. Given the demand equation we can see that country 2 has smaller consumer surplus. So, if the fixed fee needs to be determined on the basis of country 2.

The fixed fee is a function of price (P)

Fixed fee (F) = 4 × (12 – 2P) × (12 – 2P)

Fixed fee (F) = 4 × (144 – 48P + 4P²)

Fixed fee (F) = 576 – 192P + 16P²

Profit (Π) = Total Revenue – Total Cost

Π = [F + (P × (12 – P)) + (P × (12 – 2P))] – [4 × ((12 – P) + (12 – 2P))]

Π = [576 – 192P + 16P² + (12P – P²) + (12P – 2P²)] – [4 × (24 – 3P)]

Π = [576 – 192P + 16P² + (12P – P²) + (12P – 2P²)] – [4 × (24 – 3P)]

Π = [576 – 168P + 13P²] – [96 – 12P]

Π = 480 – 156P + 13P²

In order to maximize the profits we take the derivative and set it equal to zero

ΔΠ/ΔP = 26P – 156 = 0

P = 6

Fixed Fee = 576 – 192P + 16P²

Fixed Fee = 576 – (192 × 6) + (16 × 36)

Fixed Fee = 576 – 1152 + 576

Fixed Fee = 0

Q₁ = 12 – P₁ = 12 – 6 = 6

Q₂ = 12 – 2P₂ = 12 – 12 = 0

Π = 480 – 156P + 13P²

Π = 480 – (156 × 6) + (13 × 36)

Π = 480 – 936 + 468

Π = 948 – 936

Π = 12

So, if the Monopolist is not allowed to discriminate then he would prefer to go for two-part tariff as the profit is higher under two-part tariff compared to no discrimination.