Eon 365 001 problem set - in a market where demand is qp

Question

Problem Set - Answer ALL questions:

Q1. Derive and sketch functions for the following situations:

(a) There are 100 identical consumers in a market each consumer has the same demand function:

q = 10 - ½p

Find the market demand.

(b) There are 25 high demand consumers

q^H = 10 - 1/10p^H

and there are 40 low demand consumers with inverse demand

p^L = 60 - 7.5q^L

Find the market demand.

(c) Market demand is Q = 1000 - 2P

There is a competitive group of firms that supplies according to the supply function

q^F = ½p - 50

Find the demand facing a dominant firm (a firm that faces the residual demand).

(d) Market demand is Q = 100 - 1/3p. A market consists of two firms so that Q = q₁+q₂. Firm 1 produces output level q₁. Find the residual (left over) demand for Firm 2.

(e) There are 10 firms in a market. Each firm has marginal costs mc = 20 + 5q and no fixed costs. Find the industry Supply function.

Q2. Derive and sketch the marginal revenue functions for each of the demand functions from question 1 (parts a through d).

Q3. In a market where demand is Q(P) = 200 - 4/3P, a monopolist has increasing marginal costs mc = 20 + ¼q.

(a) Find the Inverse demand function for the monopolist P(Q).

(b) Write out the marginal revenue function of the monopolist.

(c) Find the profit-maximizing monopoly quantity, q^M and price, p^M.

(d) Calculate the Consumer and Producer Surplus and the Deadweight Loss of Monopoly.

Q4. Suppose that in a market with demand Q(P) = 200 - 4/3P there is a single dominant firm with marginal cost mc = 20+ ¼q and a group of competitive firms with the supply function q^f = 2p - 40.

(a) Find the residual demand function of the dominant firm.

(b) For quantities/prices where the dominant firm shares the market with the competitive firms (the top/flatter portion of residual demand), determine the marginal revenue function of the dominant firm.

(c) Find the quantity the dominant firm will produce, q^Dom, the price the firm will set in the market, p^Dom; and the quantity that will be produced by the competitive group of firms, q^F.

(d) Draw a graph that shows the equilibrium and clearly show both Producer surplus to the dominant firm and Producer surplus to the competitive firms.

(e) Calculate the surplus to the dominant firm, the competitive group of firms and the level of consumer surplus.

Q5. Individual customer demand, d_i, at an amusement park operated by a is Q = 30 - .4P and the marginal cost of each ride is mc = 5. There are 100 identical customers.

(a) Find the monopoly price, p^M and number of rides per customer, q^M_i. Calculate the profits to the monopolist. Note that the monopolist equate mr = mc where mr is based on the total demand.

(b) Find the optimal {A, p} when the monopolist charges an admission fee, A and a price per ride p. Calculate the profits to the monopolist.

Q6. A monopolist with increasing marginal costs sells a good into two markets. The demand in market 1 is q₁ = 50p₁ while the demand in market 2 is q₂ = 100 - 2/3p₂. The marginal cost function of the monopolist is c = 5 + 1/2q.

(a) If the monopolist is unable to practice 3rd degree price discrimination. Find the profit maximizing price the monopolist should set, p^M.

(b) If the monopolist is able to use third degree price discrimination draw a graph to show how the prices will be determined in each market.

(c) Find the profit maximizing price the monopolist should set with price discrimination. How much is purchased in each market?

(d) Explain whether it is possible to conclude that TS decreases with the move to third degree price discrimination.

Q7. A mobile phone service provider sells long-distance air time to two different consumers. The low demand consumer has demand function q^L = 125 - 250p^L while the high demand consumer has demand function q^H = 200 - 250p^H where quantity is measured in minutes per month and price is measured in $/minute. The marginal cost of providing the long-distance air time is $0.05/minute. The firm is unable to distinguish the two consumers. The firm will offer two different service contracts, H and L, each specifying an amount paid, T in return for a quantity of long-distance air time, q, H = {T^H, q^H} and L = {T^L, q^L}.

(a) Write out functions for the Willingness to Pay for both consumers, WTP^H and WTP^L.

(b) Write out the profit-maximizing condition the firm uses when setting the quantity q^H. Determine the optimal level of q^H. How does q^H compare to the amount that would be sold to the high demand consumer in a competitive market?

(c) Write out the profit-maximizing condition the firm uses when setting the quantity q^L when the firm wants to insure that high-demand consumers do not want to take the contract L. (incentive compatibility). Determine the optimal level of q^L. How does q_L compare to the amount that would be sold to the low demand consumer in a competitive market?

(d) Find the amounts that should be charged for q^L and q^H (T^L and T^H), and write out the contracts H and L: What is the average price ($/minute) of each user?

Q8. A company is introducing a new durable-handle, disposable-blade razor into the market (with 6 blades!). There are two types of users, daily shavers and baby-faced weekly shavers. The demand for disposable blades for a daily shaver is q^H = 93.25 - 55p while for a weekly shaver, the demand is q^L = 30.75 - 25p. The marginal cost of the disposable blade is $0.25. Suppose that the durable handles can be produced at zero marginal cost. The firm is unable to distinguish a daily shaver from a weekly shaver. It is not possible to use the blades without the handle, and the handle cannot be used with any other blades. The firm sets the price of the handle at A and the price of disposable blades at p.

(a) If the firm were to set the price of the disposable blade at p = 0.25, determine the maximum price, A they could sell the handle for and still have both users purchase. Determine the surplus (profits) to the firm from doing this.

(b) When p = 0.25 as in part (a) above, explain whether the marginal revenue from selling an additional disposable blade (increasing q) is larger or smaller than the marginal cost of producing the additional disposable blade.

(c) Find the total demand for disposable blades, Q = q^H + q^L (note: this will be a kinked demand function, it should consist of two segments). Determine the inverse demand, P(Q) over the range where both q^H and q^L are positive and write this using q^H + q^L in place of Q.

(d) Explain why the profit maximizing condition for the firm is to set q^H and q^L so that

P(q^H + q^L) + dP/dQ(q^H - q^L) = c

Determine the optimal level of q^H.

(e) Use the daily shaver demand curve to find the price the firm will charge for the disposable blade, p. Find q^L (using p and the weekly shaver demand function). Determine the price of the handle, A.

(f) Determine the surplus to the firm when they set A and p according to (e) and compare your result to the surplus in (a).

Q9. A monopolist faces no competition in a market. The individual consumer demand for the good is q = 1500 - 30p. The marginal cost of production is constant, c = $10.

(a) Find the uniform monopoly price and quantity, p^m and q^m. Calculate the profits to the firm.

(b) If the firm uses a two-block structure setting a price p₁ on the first q₁ units purchased and price p₂ on all additional units purchased. Calculate the level of profits to the firm under the block-pricing structure.

(c) If the firm uses a three-block structure setting price p₁ on the first q₁ units purchased, price p₂ on the next q₂ units purchased and price p₃ on all additional units purchased. Calculate the level of profits to the firm under this block-pricing structure.