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Suppose mountain spring water can be produced at no cost and the inverse demand for mountain spring water is P = 1200 - 0.2Q. Answer the following questions.


a. Suppose the market of mountain spring water is supplied by a monopoly firm that cannot price discriminate. Find the monopoly firm's profit-maximizing price and quantity of production. (10 pts.) [Hint: MR = P + (P/Q)*Q]


b. Suppose the market of mountain spring water is supplied by two firms (Firm A and firm B) that behave like a Cournot duopoly. Find the Nash Equilibrium price and quantity of production for each firm. (10 pts.) [Hint: Q = QA + QB; MRA = P + ((P/QA)*QA and MRB = P + ((P/QB)*QB]


c. Suppose the market of mountain spring water is supplied by two firms (firm A and firm B) that behave like a Stackelberg duopoly where firm A is the leader and firm B is the follower. Find the Nash Equilibrium price and quantity of production for each firm. (10 pts.) [Hint: Q = QA + QB; MRA = P + ((P/QA)*QA and MRB = P + ((P/QB)*QB]

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28 Sep 2019

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Related questions

Two firms compete by setting quantities. Firm A has a marginal cost of 10, while Firm B has a marginal cost of 20. The demand function is given by: p = 90 – Q/2, where Q = qA + qB. 

a. What are the equilibrium quantities if both firms set quantities simultaneously (Cournot competition)? 

b. What are the equilibrium quantities if Firm A sets quantity first (Stackelberg competition)?

Suppose that there are two firms competing in the market for taxi services. Big Ben Taxis has the marginal cost MCB = $9 per trip, and the fixed cost FCB = $3,000,000. While Whitehall Taxis has the marginal cost MCW = $15 per trip, and the fixed cost FCW = $1,000,000. Inverse demand for taxi trips in the market is given by the function, P = 75 − Q/10,000 . In this equation, P is the price of a taxi trip, and Q is the total quantity of taxi trips supplied by the two taxi companies.

Question 1: Find the equilibrium price and quantities for the case in which the two taxi companies engage in Cournot (quantity) competition. What profits will Big Ben Taxis and Whitehall Taxis earn. (11 Marks)

Question 2: Using your answers to question 1, determine which firm has the greater market power. (4 Marks)

Question 3: Now suppose that a firm can only supply taxi services if it purchases a licence from the government. What is the highest fee that the government can charge for a license, if the government wants both Big Ben Taxis and Whitehall Taxis to purchase a license? (Note: A licence does not place a limit on the number of taxi trips a company can supply. You should assume that both firms are charged the same fee.) (2 Marks)

Question 4: If, instead, the government wants to maximise the revenue it receives from taxi license fees, how many licenses should it sell, and what fee should it charge? (8 Marks)

Q1 and Q2 answered, can anyone help with Q3 and Q4?

Q1

)P = 75 - (Q / 10,000) = 75 - [(QB + QW) / 10,000] (

QB, QW: Quantity of Big Ben & Whitehall respectively)

Total revenue, Big Ben, TRB = P x QB = 75QB - [(QB2 + QW.QB) / 10,000]

Marginal revenue, Big Ben, MRB = dTRB / dQB = 75 - [(2QB + QW) / 10,000]

Equating MRB with MCB, 75 - [(2QB + QW) / 10,000] = 9 750,000 - 2QB - QW = 90,000 2QB + QW = 660,000 .......(1) [Reaction function of Big Ben]

Total revenue, Whitewall, TRW = P x QW = 75QW - [(QW.QB + QW2) / 10,000]

Marginal revenue, Whitehall, MRW = dTRW / dQW = 75 - [(QB + 2QW) / 10,000]

Equating with MCW, 75 - [(QB + 2QW) / 10,000] = 15 750,000 - QB - 2QW = 150,000 QB + 2QW = 600,000 .....(2) [Reaction function of Whitehall]

Cournot equilibrium is obtained by solving (1) and (2).

2QB + QW = 660,000 .......(1) (2) x 2 results: 2QB + 4QW = 1,200,000 ...(3)

(3) - (1) yields:

3QW = 540,000 QW = 180,000 QB = 600,000 - 2QW = 600,000 - (2 x 180,000) = 600,000 - 360,000 = 240,000

Q = QW + QB = 180,000 + 240,000 = 420,000 P = 75 - (420,000 / 10,000) = 75 - 42 = 33

Profit = Total revenue - Total cost = (P x Q) - [FC + (MC x Q)] = Q x (P - MC) - FC [FC: Fixed cost]

Profit, Big Ben = [240,000 x (33 - 9)] - 3,000,000 = (240,000 x 24) - 3,000,000 = 5,760,000 - 3,000,000 = 2,760,000

Profit, Whitehall = [180,000 x (33 - 15) - 1,000,000 = (180,000 x 18) - 1,000,000 = 3,240,000 - 1,000,000 = 2,240,000 (3)

The fixed fee charged by government is equal to consumer surplus (CS) for each company, where CS is the area between demand curve and market price.

From demand curve, When Q = 0, P = 75 [Reservation price] For Big Ben, CS = (1/2) x $(75 - 33) x 240,000 = (1/2) x $42 x 240,000 = $5,040,000

This is the maximum license fee for Big Ben. For Whitehall, CS= (1/2) x $(75 - 33) x 180,000 = (1/2) x $42 x 180,000 = $3,780,000

This is the maximum license fee for Whitehall. (4) Revenue is maximized when marginal revenue is zero.

Marginal revenue, Big Ben, MRB = dTRB / dQB = 75 - [(2QB + QW) / 10,000] So, 75 - [(2QB + QW) / 10,000] = 0 750,000 - 2QB - QW = 0 2QB + QW = 750,000 ......(4)

Marginal revenue, Whitehall, MRW = dTRW / dQW = 75 - [(QB + 2QW) / 10,000] So, 75 - [(QB + 2QW) / 10,000] = 0 750,000 - QB - 2QW = 0 QB + 2QW = 750,000 ....(5) Multiplying (5) by 2, 2QB + 4QW = 1,500,000 ......(6) 2QB + QW = 750,000 ...... (4) (6) - (4) gives: 3QW = 750,000 QW = 250,000 QB = 750,000 - (2 x 250,000) = 750,000 - 500,000 = 250,000 Q = QB + QW = 250,000 + 250,000 = 500,000 P = 75 - (500,000 / 10,000) = 75 - 50 = $25

2] Two firms, A and B, behave as Bertrand competitors. Their products are differentiated, leading to the following demand and total cost equations:

Firm A Demand: QA = 200  . PA + PB Firm A Total Cost: TCA = 50·QA

Firm B Demand: QB = 200 .  PB + PA Firm B Total Cost: TCB = 50·QB

In this situation, the price-reaction functions become:

Firm A: PA = 125 + 0.5PB Firm B: PB = 125 + 0.5PA

With respect to the firm demand equations, explain the rationale for (i) the increase in own price leading to a decrease in the firm's own quantity and (ii) the increase in the other firm's price increasing the firm's own quantity. Find the Bertrand/Nash equilibrium prices. Determine each firm's profit at the Bertrand prices.

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