Strategic Output Decisions in Duopoly and Monopoly Markets

Posted on Dec 26, 2024 in Economy

Consider a market for a homogeneous product where the market demand is described by the equation: P = 600 − 3Q, where Q represents the total quantity measured in pounds per day, and P the price per unit for the product measured in dollars. There are two firms — Firm A and Firm B — in this market, and in different scenarios their marginal revenue functions are described by the equations: MR = 600 − 6Q, MRA = 600 − 6QA − 3QB, MRB = 600 − 3QA − 6QB, MRSA = 375 − 3QA, and MRSB = 375 − 3QB, where S represents Stackelberg, A represents Firm A, and B represents Firm B. Each firm produces at a constant marginal cost of $150 and has no fixed costs.

Cournot Duopoly

Strategically, if each firm makes its output decision competitively and simultaneously, then Firm A’s optimal level of output is determined as follows: For Firm A, solve MRA = MCA, that is 600 − 6QA − 3QB = 150, and find Firm A’s reaction function: QA = 75 − 0.5QB. Similarly, for Firm B, solve MRB = MCB, that is 600 − 3QA − 6QB = 150, and find Firm B’s reaction function: QB = 75 − 0.5QA. Solve the reaction functions simultaneously, that is solve QA = 75 − 0.5(75 − 0.5QA), and find QA = 50. Substituting QA = 50 into the reaction function for QB, solve QB = 75 − (0.5 × 50) and find QB = 50. Q = q1+q2. MR = 600 – 6Q. Once you find q1 and q2, these are the optimal levels of output for each firm. We need total Q now which is Q=q1+q2 = 50 + 50 = 100 and P = 600 – 3(Q) = 300. This is the price firm A and B charge. Profit will be Pro = TR – VC – FC = P(q1) – MC1(q1) – FC which is 0 here.

Stackelberg Duopoly

Strategically, if Firm A makes its output decision first and Firm B follows, then Firm A’s optimal level of output is determined as follows: If Firm A is the leader, then firm B is the follower and both firms are involved in a Stackelberg duopoly. For Firm B, solve MRB = MCB, that is 600 − 3QA − 6QB = 150, and find Firm B’s reaction function: QB = 75 − 0.5QA. This is how Firm B will react to any output decision Firm A makes. Since Firm A is the leader, solve for Firm A, MRSA = MCA, that is 375 − 3QA = 150, and find QA = 75. Firm A’s output is 75 pounds. Using Firm B’s reaction function, solve QB = 75 − (0.5 × 75) and find QB = 37.5. These are the optimal levels of output. Now we find Q=q1+q2 = 112.5, P = 600 – 3Q = 262.5, which is the price firms A and B charge. Find each individual profit Pro = P(q1) – MC1(q1), same for 2, and that is profit.

Monopoly Output

Strategically, if both firms collude when making their output decisions, then Firm A’s optimal level of output is determined as follows: Monopoly output Q=qa+qb and P = 600 – 3Q and MR = 600 – 6Q. Now set MR=MC, 600 – 6Q = 150, solve for big Q. In this case of collusion qa=qb so Q=q1+q2 = 2q1=2q2 and these q are the optimal levels of output. Find P = 600 – 3(Q) from MR=MC and find P, which is price firms charge. Profit is the same for both = TR – VC – FC = PQ – (MC)P.

Case Study: The Montague and The Capulet Hotels

Consider that in Positano, a coastal town, Jennifer Montague and Randy Capulet are each owners of the only two all-inclusive hotels — The Montague and The Capulet. At each hotel, suites are rented on a weekly basis only, and the different sizes and years of establishment between the rival hotels have led to Randy making his decision about the number of suites rented after observing Jennifer’s decision about the number of suites rented. The Capulet has a fixed cost of $115,000 and a constant marginal cost of $6000 while the Montague has a fixed cost of $485,000 and a constant marginal cost of $4000. The market demand for a suite is described by the equation: P = 16000 − 4Q, where Q represents the total quantity of suites rented per year, and P the (weekly) price per suite measured in dollars. As the only providers of an all-inclusive hotel, the marginal revenue functions for the Montague and the Capulet in different scenarios are described by the equations: MR = 16000 − 8Q, MRC = 16000 − 4QM − 8QC, MRM = 16000 − 8QM − 4QC, MRSC = 9000 − 4QC, and MRSM = 11000 − 4QM, where S represents Stackelberg, C represents the Capulet hotel, and M represents the Montague hotel.

Lerner Index Calculation

When MC is constant = AVC. At its optimal level of suites rented, Capulet’s Lerner index is calculated as follows: At its optimal level of output, the Capulet charges a price of $7500 and enjoys a marginal cost $6000. Recall that for the Lerner index, L = (P − MC) ÷ P. Solve (7500 − 6000) ÷ 7500 and find that the Capulet’s Lerner index is 0.2. Similarly, at its optimal level of output, the Montague charges a price of $7500 and enjoys a marginal cost $4000. Solve (7500 − 4000) ÷ 7500 and find that the Montague’s Lerner index is 0.47 (rounded to two decimal places).

Price Elasticity of Demand

At the current market price, the price elasticity of demand for Capulet’s hotel suites is calculated as follows: At its optimal level of output, the Capulet charges a price of $7500 and enjoys a marginal cost $6000. Recall that for the Lerner index, L = (P − MC) ÷ P. Solve (7500 − 6000) ÷ 7500 and find that the Capulet’s Lerner index is 0.2. Recall that the price elasticity of demand can be calculated as PED = − (1 ÷ L). Solve − (1 ÷ 0.2) and find that the price elasticity of demand for a suite at the Capulet is −5. Also, at the current market price, P = $7500, the quantity demanded (and traded) is 2125 suites. Recall that the market level price elasticity of demand can be calculated as PED = (1 ÷ slope of the demand curve) × (P ÷ Q). Solve (1 ÷ −4) × (7500 ÷ 2125) and find that the market level price elasticity of demand is −0.8824. Greater absolute value of PED is more sensitive to change.

Comparison of Oligopoly Models

Comparing the oligopoly models (from lowest value to highest value):

  • Individual quantity: Collusion, Stackelberg (follower), Cournot, Stackelberg (leader), Bertrand
  • Market quantity: Collusion, Cournot, Stackelberg, Bertrand
  • Market price: Bertrand, Stackelberg, Cournot, Collusion
  • Individual profit: Bertrand, Stackelberg (follower), Cournot, Stackelberg (leader), Collusion

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Case Study: Lettermen Lawns Services

Consider a mid-sized town where Lettermen Lawns is the only firm that provides garden and lawn care services. As a well established firm, Lettermen Lawns faces a constant marginal cost of $20 and no fixed cost. The market demand for garden and lawn care services is described by the equation: P = 100 − 10Q, where P is the price charged and Q is the quantity of gardens and lawns serviced per day. As the only provider, Lettermen Lawns’ marginal revenue is described by the equation: MR = 100 − 20Q.

Single Price Strategy

By charging a single price, the optimal quantity will occur where MR = MC. Solve 100 – 20Q = 20 and find Q = 4. Lettermen Lawns will service 4 gardens and/or lawns per day. Using the equation that represents the market demand, solve P = 100 – (10 × 4) and find P = 60. The firm’s optimal price is $60. Solve $60 × 4 to find total revenue = $240. Also, solve $20 × 4 to find total variable cost = $80. With fixed cost = $0, solve profit = 240 – 80 – 0 and find that the firm’s profit is $160.

If Lettermen Lawns charges a single price per service to all customers, then the total consumer surplus at the optimal price is calculated as follows: Using the price = $60 and the quantity = 4, solve CS = 0.5 × ($100 − $60) × (4) and find CS = $80. The total consumer surplus in the market is $80. To find the efficient quantity, solve 100 – 10Q = 20 and find Q = 8. Using the price = $60 and marginal cost = $20 at quantity = 4, solve 0.5 × ($100 − $60) × (8 − 4) and find DWL = $80. The total amount of inefficiency (the total deadweight loss) is $80.

First-Degree Price Discrimination

If Lettermen Lawns charges each customer the price they are willing to pay for each unit of service bought, then the optimal amount of gardens and lawn serviced each day is calculated as follows: By charging each customer the price they are willing to pay for each unit of service bought, the firm is engaging in first-degree price discrimination. The optimal quantity will occur where P = MC. Solve 100 – 10Q = 20 and find Q = 8. Lettermen Lawns will service 8 gardens and lawns. The price charged will be a range of prices from $100 to $20. Solve [0.5 × (100 – 20) × 8] + [20 × 8] to find total revenue = $480. Also, solve 20 × 8 to find total variable cost = $160. With fixed cost = $0, solve profit = 480 – 160 – 0 and find that the firm’s profit is $320.

If Lettermen Lawns charges each customer the price they are willing to pay for each unit of service bought, then the total consumer surplus at the optimal amount of gardens and lawn serviced each day is calculated as follows: By charging each customer the price they are willing to pay for each unit of service bought, the firm is engaging in first-degree price discrimination. The optimal quantity will occur where P = MC. Solve 100 – 10Q = 20 and find Q = 8. Since consumers are paying the price they are willing to pay, then the total consumer surplus in the market is $0. Since the firm is trading the quantity where P = MC, the total amount of inefficiency (the total deadweight loss) is $0.

Price Discrimination Based on Time of Service

Some customers prefer having their lawns and gardens serviced earlier in the day. If Lettermen Lawns charges $80 for the gardens and lawns serviced earlier in the day, and then $60 for the remaining gardens and lawns serviced later in the day, then the optimal amount of gardens and lawns serviced each day is calculated as follows: At P = $80, solve 80 = 100 – 10Q and find Q = 2. At P = $60, solve 60 = 100 – 10Q and find Q = 4. The total number of gardens/lawns serviced each day is 4. Two gardens and/or lawns are serviced at a price of $80, and the remaining 2 gardens and/or lawns are serviced at a price of $60. Solve ($80 × 2) + ($60 × 2) to find total revenue = $280. Also, solve $20 × 4 to find total variable cost = $80. With fixed cost = $0, solve profit = 280 – 80 – 0 and find that the firm’s profit is $200.

Some customers prefer having their lawns and gardens serviced earlier in the day. If Lettermen Lawns charges $80 for the gardens and lawns serviced earlier in the day, and then $60 for the remaining gardens and lawns serviced later in the day, then the total consumer surplus at the optimal amount of gardens and lawns serviced each day is calculated as follows: At P = $80, solve 80 = 100 – 10Q and find Q = 2. At P = $60, solve 60 = 100 – 10Q and find Q = 4. The total number of gardens and lawns serviced each day is 4. Two gardens and lawns are serviced at a price of $80, and the remaining 2 gardens and lawns are serviced at a price of $60. Using these prices and quantities, solve CS = [0.5 × ($100 − $80) × (2)] + [0.5 × ($80 − $60) × (2)] and find CS = $40. The total consumer surplus in the market is $40. To find the efficient quantity, solve 100 – 10Q = 20 and find Q = 8. Using the price = $60 and marginal cost = $20 at quantity = 4, solve 0.5 × ($100 − $60) × (8 − 4) and find DWL = $80. The total amount of inefficiency (the total deadweight loss) is $80.