CHAPTER 9

OLIGOPOLY AND FIRM ARCHITECTURE

1. The demand function for a product sold by an oligopolist is given below:

QD = 370 – P

The firm’s marginal cost function is given below:

MC = 10 + 4Q

Calculate the equilibrium price and quantity.

Solution:

P = 370 – Q so TR = 370Q – Q2 and MR = 370 – 2Q

MR = 370 – 2Q = 10 + 4Q = MC so Q = 60 and P = 310

2. The demand function for a product sold by an oligopolist is given below:

QD = 135 – 0.5P

The firm’s marginal cost function is given below:

MC = 30 + 4Q

Calculate the equilibrium price and quantity.

Solution:

P = 270 – 2Q so TR = 270Q – 2Q2 and MR = 270 – 4Q

MR = 270 – 4Q = 30 + 4Q = MC so Q = 30 and P = 210

3. An oligopolist is currently charging a price of $600 and is selling 300 units of output per day. If the firm increases price above $600, then quantity demanded will decline by 3 units for every $1 increase in price. If the firm reduces price below $600, then the quantity demanded will increase by 1.5 units for every $1 decrease in price. If the firm's marginal cost curve is horizontal, within what range could marginal cost vary without giving the firm an incentive to change price or quantity?

Solution:

The kinked demand curve model applies here. At prices above $600, the demand and marginal revenue functions are:

Q = 2100 – 3P so P = 700 – (1/3)Q and MR = 700 – (2/3)Q

At prices below $300, the demand and marginal revenue functions are:

Q = 1200 – 1.5P so P = 800 – (2/3)Q and MR = 800 – (4/3)Q

The firm will continue to charge a price of $600 so long as the optimal level of output is 300 units. Q = 300 will be optimal so long as MC is between the two values of MR located at the kink. Thus, MR = 700 – (2/3)(300) = 500 is the upper limit of MC and MC = 800 – (4/3)(300) = 400 is the lower limit of MC.

4. An oligopolist is currently charging a price of $100 and is selling 400 units of output per day. If the firm increases price above $100, then quantity demanded will decline by 40 units for every $1 increase in price. If the firm reduces price below $100, then the quantity demanded will increase by 10 units for every $1 decrease in price. If the firm's marginal cost curve is horizontal, within what range could marginal cost vary without giving the firm an incentive to change price or quantity?

Solution:

The kinked demand curve model applies here. At prices above $100, the demand and marginal revenue functions are:

Q = 4400 – 40P so P = 110 – 0.025Q and MR = 120 – 0.05Q

At prices below $100, the demand and marginal revenue functions are:

Q = 1400 – 10P so P = 140 – 0.10Q and MR = 140 – 0.20Q

The firm will continue to charge a price of $100 so long as the optimal level of output is 400 units. Q = 400 will be optimal so long as MC is between the two values of MR located at the kink. Thus, MR = 110 – (0.05)(400) = 90 is the upper limit of MC and MC = 140 – (0.20)(400) = 60 is the lower limit of MC.

5. Firms A and B operate as a centralized cartel. Their marginal cost functions are defined below:

MCA = 2000 + 25QA MCB = 2000 + 6.25QB

The firms face the following market demand curve:

Q = 1000 – 0.05P

Determine the market price that the firms should charge and the quantity of output that should be produced by each firm.

Solution:

The horizontal sum of the two marginal cost functions is calculated as follows:

QA = –80 + 0.04MCA QB = –320 + 0.16MCB

QA + QB = Q = –400 + 0.20MC MC = 2000 + 5Q

Equilibrium output is derived by setting MC = MR as follows:

P = 20000 – 20Q so MR = 20000 – 40Q

MC = 2000 + 5Q = 20000 – 40Q = MR so Q = 400, P = 12000, and MC = 4000

Substituting the equilibrium MC into the individual firm's MC functions yields:

QA = –80 + (0.04)(4000) = 80 and QB = –320 + (0.16)(4000) = 320

6. Firms A and B operate as a centralized cartel. Their marginal cost functions are defined below:

MCA = 25QA MCB = 6.25QB

The firms face the following market demand curve:

Q = 1000 – 0.10P

Determine the market price that the firms should charge and the quantity of output that should be produced by each firm.

Solution:

The horizontal sum of the two marginal cost functions is calculated as follows:

QA = 0.04MCA QB = 0.16MCB

QA + QB = Q = 0.20MC MC = 5Q

Equilibrium output is derived by setting MC = MR as follows:

P = 10000 – 10Q so MR = 10000 – 20Q

MC = 5Q = 10000 – 20Q = MR so Q = 400, P = 6000, and MC = 2000

Substituting the equilibrium MC into the individual firm's MC functions yields:

QA = (0.04)(2000) = 80 and QB = (0.16)(2000) = 320

7. A market that follows the price leadership of a barometric firm has the following demand function:

QDM = 1400 – 2P

The follower firms have the following aggregate marginal cost function:

MCF = 100 + 0.50QMC

The barometric firm has a horizontal marginal cost curve equal to $300. Determine total industry output, market price, and the division of output between the barometric firm and the follower firms.

Solution:

The barometric firm's demand function (QDL) is determined by taking the horizontal difference between the market demand function and the followers' aggregate marginal cost function when P = MCF as follows:

QMC = 2MCF – 200 so QDL = QDM – QMC = 1400 – 2P – 2P + 200 = 1600 – 4P

The barometric firm's marginal revenue function is:

P = 400 – 0.25QDL so MR = 400 – 0.5QDL

The barometric firm will produce the quantity of output that equates its marginal cost with the marginal revenue associated with its demand function:

MC = 300 = 400 – 0.5QDL = MR so QDL = 200 and P = 350

The follower firms take the price determined by the barometric firm as given and produce a quantity that equates their aggregate marginal cost with price:

P = 350 = 100 + 0.5QF = MC so QF = 500 and total output is 700.

8. A market that follows the price leadership of a barometric firm has the following demand function:

QDM = 800 – 0.50P

The follower firms have the following aggregate marginal cost function:

MCF = 2QMC

The barometric firm has a horizontal marginal cost curve equal to $600. Determine total industry output, market price, and the division of output between the barometric firm and the follower firms.

Solution:

The barometric firm's demand function (QDL) is determined by taking the horizontal difference between the market demand function and the followers' aggregate marginal cost function when P = MCF as follows:

QMC = 0.50MCF so QDL = QDM – QMC = 800 – 0.50P – 0.50P = 800 – P

The barometric firm's marginal revenue function is:

P = 800 – QDL so MR = 800 – 2QDL

The barometric firm will produce the quantity of output that equates its marginal cost with the marginal revenue associated with its demand function:

MC = 600 = 800 – 2QDL = MR so QDL = 100 and P = 700

The follower firms take the price determined by the barometric firm as given and produce a quantity that equates their aggregate marginal cost with price:

P = 700 = 2QF = MC so QF = 350 and total output is 450.

9. Two rival companies sell software packages that are perfect substitutes. The software is sold over the web as a download, so the marginal cost is zero. The demand for the software is Q = 1000 – P. Assume that these firms are Cournot duopolists. Derive the reaction functions for the two firms, the quantity each will produce, and the market price that will be charged.

Solution:

Derive the marginal revenue function for each firm and set it equal to zero as follows:

MR1 = 1000 – Q2 – 2Q1 = 0

MR2 = 1000 – Q1 – 2Q2 = 0

Reaction function for firm 1: Q1 = (1000 – Q2)/2 = 500 – 0.50 Q2

Reaction function for firm 2: Q2 = (1000 – Q1)/2 = 500 – 0.50 Q1

Solve the reaction functions simultaneously for the quantities produced:

Q1 = 500 – (0.50)(500 – 0.50 Q1) = 250 + 0.25 Q1 => Q1 = 250/0.75 = 333.33

Q2 = 500 – (0.50)(333.33) = 333.33

Market price is found by substituting quantity into the demand function:

P = 1000 – (Q1 + Q2) = 1000 – 666.67 = $333.33

10. Two rival companies sell software packages that are perfect substitutes. The software is sold over the web as a download, so the marginal cost is zero. The demand for the software is Q = 900 – P. Assume that these firms are Cournot duopolists. Derive the reaction functions for the two firms, the quantity each will produce, and the market price that will be charged.

Solution:

Derive the marginal revenue function for each firm and set it equal to zero as follows:

MR1 = 900 – Q2 – 2Q1 = 0

MR2 = 900 – Q1 – 2Q2 = 0

Reaction function for firm 1: Q1 = (900 – Q2)/2 = 450 – 0.50 Q2

Reaction function for firm 2: Q2 = (900 – Q1)/2 = 450 – 0.50 Q1

Solve the reaction functions simultaneously for the quantities produced:

Q1 = 450 – (0.50)(450 – 0.50 Q1) = 225 + 0.25 Q1 => Q1 = 225/0.75 = 300

Q2 = 450 – (0.50)(300) = 300

Market price is found by substituting quantity into the demand function:

P = 900 – (Q1 + Q2) = 900 – 600 = $300