1. (1DPD) Imagine there are ten different consumers in the market and each has individual demand curve given by P_i = i - q, where i represents consumer number (i={1..10}).

That is:

a. the first consumer has a demand curve of P₁ = 1 - q

b. the second consumer has a demand curve of P₂ = 2 - q

c. the third consumer has a demand curve of P₃ = 3 - q, etc.

Assume that this market is served by a monopolist with a total cost function of C(Q) = Q/2, where Q is aggregate demand (i.e. Q = q₁ + q₂ + ... + q₁₀). Firm's marginal cost is therefore MC = . Also assume that monopolist can perfectly price discriminate using first-degree price (1DPD) discrimination methods.

a. Draw the individual demand and marginal cost of the producer. You should have 10 different demand curves for each consumer. Do not forget to appropriately label your graphs.

b. The monopolist decides to use the block-pricing method of 1DPD. Calculate the price of "package" for each consumer. What quantity would the monopolist include in each package?

c. The monopolist decides to use two-part pricing (two-part tariff) method of 1DPD. The monopolist chooses a fixed component T - tariff (which consumers must pay for the right to buy any amount of the good), plus a variable component p_i (which consumers pay per unit for the goods). Calculate the price of tariff (T) and per unit price (p_i) for each consumer.

d, Calculate firm's profit under the 1DPD. Note that the profit is the same under the two-part tariff and block-pricing methods.

e, Calculate consumer surplus to each buyer. Note that the consumer surplus is same under the two-part tariff and block-pricing methods.

2. (1DPD) A monopolist faces the inverse market demand curve P = 6 - Q, and the total cost function C = 2Q. The firm's marginal cost is therefore MC = 2. Assume the demand curve is made up of many consumers each with unit demand for this good (i.e. each consumer will either buy one or nothing and the market demand curve is the aggregation of all these unit-demand consumers.)

• a, the monopolist cannot price discriminate, find the monopolist's price, quantity, profit, and consumer surplus.
• b, What assumptions are necessary for the monopoly to be able to perfectly price discriminate (1DPD)?
• c, If the monopolist can perfectly price discriminate, solve for its profits, and total consumer surplus. (Hint: since consumers have unit demand, all three methods - separate prices per unit, package prices, or a two-part tariff are trivially the same. For example, the "package" contains exactly one good, and every consumer receives a price equal to her valuation.)

3. (1DPD) Assume there are only 2 consumers in the market, and both have an individual demand for a product given by: P₁ = 6 - q₁ and P₂ = 6 - q₂. A monopolist firm that services both buyers, has a marginal cost of MC = 2.

• a. Show that inverse market demand function is P = 6 - Q/2 (where, Q = q₁ + q₂)
• b. Using the market demand, derive marginal revenue of the monopolist.
• c. Plot the market demand, marginal revenue, and marginal cost curves.
• d. Calculate the monopolist's profit maximizing price, quantity, and profit.
• e. Now assume that monopolist is using 1DPD (block-pricing). How much will each buyer pay for the package? How many units will be included in each package?

4. (3DPD) Suppose that a local monopolist car rental company estimates that the student demand for its car is: P_S = 100 - q_S/2 and non-student customer demand is given by: P_NS = 150 - q_NS/2. The marginal cost of renting out a car is estimated to be $25per car.

a, Assume that the monopolist cannot differentiate individuals in these two groups. Monopolist is thinking of using uniform pricing to meet the demands of both groups. That is, to charge single profit maximizing price for both groups.

• i. Derive inverse total market demand (Hint: Total market demand Q = q_S + q_NS). Clearly state the market price range to serve both the student and non-student population.
• ii. Use the inverse demand in part i) and derive marginal revenue.
• iii, Calculate profit maximizing price and quantity. Does the price satisfy the range you defined in part i)?
• iv, Calculate the profit of the monopolist.

b. Now assume that monopolist is thinking about third-degree price discrimination (3DPD).

• i, What price should this firm charge students and what price should it charge non-students? Find profit maximizing quantity under both scenarios. ii.Calculate the consumer surplus of two types of buyers