Two Firms Compete In A Market To Sell A Homogeneous Product
Two Firms Compete In A Market To Sell A Homogeneous Product With In
This assignment encompasses three core economic analysis scenarios involving different market structures and strategic interactions among firms. The first scenario compares the behavior and outcomes—output levels and profits—of two firms competing under various market models: Bertrand, Cournot, Stackelberg, and collusion. The second scenario evaluates the impact of a technological innovation on a firm's production strategy and profitability within an oligopolistic market. The third scenario involves a Cournot duopoly where firms select output levels based on best response functions, and extends into a Stackelberg leadership model, analyzing equilibrium prices, outputs, and profits. These analyses apply fundamental principles of microeconomic theory, including rivalry models, market equilibrium, and strategic decision-making, within the context of homogeneous product markets. This comprehensive examination utilizes demand functions, cost structures, and strategic assumptions to compare outcomes across different competitive frameworks.
Paper For Above instruction
Understanding the strategic interactions between firms in oligopolistic markets is fundamental in industrial organization economics. The four models—Bertrand, Cournot, Stackelberg, and collusion—each represent distinct assumptions about how firms compete and the resulting market outcomes, particularly concerning prices, outputs, and profits. Analyzing these models using the provided demand functions, costs, and strategic assumptions reveals how different competitive behaviors influence market efficiency and firm profitability.
Comparison of Competitive Models: Outputs and Profits
The first scenario involves two firms competing in a homogeneous product market described by the inverse demand function P = 600 - 3Q, where Q = Q₁ + Q₂. Each firm has constant marginal costs of $300 and no fixed costs. To compare the outcomes under different competition models, we analyze each case individually:
1. Bertrand Competition
The Bertrand model assumes firms set prices simultaneously and competitively to undercut each other until price equals marginal cost. Given both firms share the same costs of $300 and face a demand at P = 600 - 3Q, the equilibrium price in Bertrand competition will align with marginal cost, i.e., P = $300. At this price, quantity demanded is:
Q = (600 - P) / 3 = (600 - 300) / 3 = 300 / 3 = 100 units
Since both firms produce at marginal cost and sell at P = $300, each firm captures half of the total quantity (assuming symmetric duopoly): Q₁=Q₂=50 units. The profit for each firm is:
π = (P - MC) Q = (300 - 300) 50 = 0
Thus, in Bertrand equilibrium, both firms earn zero economic profits, and the market price is driven down to marginal cost, leading to allocative efficiency but zero profits.
2. Cournot Competition
In Cournot competition, firms choose quantities simultaneously, assuming the other firm’s output is fixed. To derive the Cournot equilibrium, we formulate each firm's best response function. The inverse demand function is:
P = 600 - 3Q, where Q = Q₁ + Q₂
Each firm’s revenue:
R_i = P Q_i = (600 - 3(Q_i + Q_j)) Q_i
Cost functions:
C_i = 300 * Q_i
Profit functions:
π_i = R_i - C_i = (600 - 3(Q_i + Q_j)) * Q_i - 300 Q_i
Maximizing profit with respect to Q_i:
dπ_i/dQ_i = 600 - 3(Q_i + Q_j) - 3 Q_i - 300 = 0
Simplify:
600 - 3 Q_i - 3 Q_j - 3 Q_i - 300 = 0
300 - 6 Q_i - 3 Q_j = 0
Rearranged as the best response function for Firm 1:
Q₁ = (300 - 3 Q₂)/6 = 50 - 0.5 Q₂
Similarly, for Firm 2:
Q₂ = 50 - 0.5 Q₁
Solving these simultaneously yields the Cournot equilibrium quantities:
Q₁ = 50 - 0.5 Q₂
Q₂ = 50 - 0.5 Q₁
Substitute Q₂ into Q₁’s response:
Q₁ = 50 - 0.5 (50 - 0.5 Q₁) = 50 - 25 + 0.25 Q₁
Thus:
Q₁ - 0.25 Q₁ = 25
0.75 Q₁ = 25
Q₁ = 25 / 0.75 ≈ 33.33 units
Q₂ = 50 - 0.5 (33.33) ≈ 33.33 units
Total quantity Q ≈ 66.66 units, and the equilibrium market price is:
P = 600 - 3 * 66.66 ≈ 600 - 200 ≈ $400
The profit per firm:
π_i = (P - MC) Q_i = (400 - 300) 33.33 ≈ $1,111
3. Stackelberg Competition
The Stackelberg model frames one firm as the leader, choosing output first, with the follower reacting optimally. Suppose Firm 1 acts as the leader; it anticipates Firm 2’s best response, which derived earlier as:
Q₂ = 50 - 0.5 Q₁
Firm 1 maximizes its profit:
π₁ = (P - 300) Q₁ = (600 - 3(Q₁ + Q₂)) Q₁
Substituting Q₂:
π₁ = (600 - 3(Q₁ + 50 - 0.5 Q₁)) Q₁ = (600 - 3(50 + 0.5 Q₁)) Q₁ = (600 - 150 - 1.5 Q₁) * Q₁
Simplify:
π₁ = (450 - 1.5 Q₁) * Q₁ = 450 Q₁ - 1.5 Q₁²
Differentiate with respect to Q₁:
dπ₁/dQ₁ = 450 - 3 Q₁ = 0
Q₁ = 150 units
Plug back into the reaction function of Firm 2
Q₂ = 50 - 0.5 * 150 = 50 - 75 = -25 units
Since negative output isn’t feasible, the optimal output is constrained at zero. Hence, the Stackelberg leader’s output is limited by the follower’s response, and the market outcome becomes more complex. Adjusting for non-negativity, the equilibrium outputs are Q₁=0, Q₂=50, leading to a market price of:
P = 600 - 3(0 + 50) = 600 - 150 = $450
Firms’ profits are then calculated accordingly:
Firm 1’s profit: (450 - 300) * 0 = $0
Firm 2’s profit: (450 - 300) * 50 = $7,500
4. Collusive Behavior
In collusion, both firms act as a monopolist, setting total output to maximize joint profits, sharing the market equally. The joint profit maximization involves choosing total Q to maximize:
π_total = (P - MC) Q = (600 - 3Q - 300) Q = (300 - 3Q) * Q
Maximize π_total with respect to Q:
dπ_total/dQ = 300 - 6 Q = 0
Q = 50 units
Price in this scenario:
P = 600 - 3(50) = 600 - 150 = $450
Each firm shares equally, Q₁=Q₂=25 units, with profits:
π_i = (450 - 300) * 25 = $3,750
The collusive outcome yields higher profits than non-cooperative models, illustrating the potential efficiency loss in competitive markets absent collusion.
Impact of Technological Innovation on Firm Strategy
In the second case, BlackSpot Computers is poised to substantially reduce its marginal cost from $800 to $500 due to technological advancements. The market demand is P=5,900 - Q, with Q = Q₁ + Q₂, and both firms initially produce at costs of $800, earning revenues of $4.25 million and a net profit of $890,000. The decrease in marginal cost to $500 has significant strategic implications.
Prior to the technological change, BlackSpot’s profit was derived from the equilibrium output and price determined under the existing conditions. With the cost reduction, BlackSpot can now profitably produce and sell more units at a lower price, intensifying competition and potentially increasing market share.
By producing at the new marginal cost, BlackSpot can reduce prices to attract more customers, as profits per unit increase. The competitive environment suggests that BlackSpot would increase output to exploit the cost advantage. The increase in output not only boosts revenues but also enhances profit margins because fixed costs are already covered, and variable costs are reduced.
Calculating the potential impact involves analyzing new profit maximization where BlackSpot’s marginal cost is now $500. The new equilibrium price is constrained by market demand and Rival’s strategies. Assuming BlackSpot as a price-setter or follower in a Bertrand framework, it could set prices close to $500 to undercut competitors or match the new lower cost point. The resulting increase in output levels should substantially improve profits, possibly surpassing the initial $890,000 net profit, depending on market elasticity and competitive responses.
In particular, the reduction in marginal costs allows BlackSpot to lower its prices and/or increase output without sacrificing profitability, potentially profitable enough to displace competitors. Increased sales volume and lower costs directly improve bottom-line results, possibly doubling or tripling profits depending on market demand elasticity and competitive responses.
Output and Profit Maximization in a Cournot Setting with Asymmetric Costs
The third scenario involves two firms with different cost functions, TC₁ = 3Q₁ and TC₂=5Q₂, operating in a market with demand P = 145 - 5(Q₁ + Q₂). They decide on outputs Q₁, Q₂ assuming the other’s output is fixed (Cournot model). The responses and equilibria involve deriving each firm’s best response functions, solving for equilibrium quantities and prices, and analyzing profit outcomes.
Part A: Best Response Functions
Each firm maximizes profit:
π₁ = P Q₁ - TC₁ = (145 - 5(Q₁ + Q₂)) Q₁ - 3 Q₁
π₂ = (145 - 5(Q₁ + Q₂)) Q₂ - 5 Q₂
For Firm 1:
π₁ = (145 - 5 Q₁ - 5 Q₂) Q₁ - 3 Q₁ = (145 Q₁ - 5 Q₁² - 5 Q₁ Q₂) - 3 Q₁
Derivative w.r.t. Q₁:
dπ₁/dQ₁ = 145 - 10 Q₁ - 5 Q₂ - 3 = 0
Rearranged:
142 - 10 Q₁ - 5 Q₂ = 0
Q₁ = (142 - 5 Q₂) / 10
For Firm 2:
π₂ = (145 - 5 Q₁ - 5 Q₂) Q₂ - 5 Q₂ = (145 Q₂ - 5 Q₁ Q₂ - 5 Q₂²) - 5 Q₂
Derivative w.r.t. Q₂:
dπ₂/dQ₂ = 145 - 5 Q₁ - 10 Q₂ - 5 = 0
Rearranged:
140 - 5 Q₁ - 10 Q₂ = 0
Q₂ = (140 - 5 Q₁) / 10
Express Q₁ and Q₂ in response:
Q₁ = (142 - 5 Q₂) / 10
Q₂ = (140 - 5 Q₁) / 10
Substitute Q₂ into Q₁’s response:
Q₁ = (142 - 5 * ((140 - 5 Q₁)/10)) / 10
Calculate numerator:
142 - (5 (140 - 5 Q₁)/10) = 142 - ( (5 140) - (5 * 5 Q₁) ) / 10 = 142 - (700 - 25 Q₁)/10
Rewrite numerator:
(142 * 10) - (700 - 25 Q₁) = 1420 - 700 + 25 Q₁ = 720 + 25 Q₁
Now, Q₁ = (720 + 25 Q₁) / 100
Rearranged:
100 Q₁ = 720 + 25 Q₁
75 Q₁ = 720
Q₁ = 720 / 75 ≈ 9.6 units
Calculate Q₂:
Q₂ = (140 - 5 * 9.6)/10 ≈ (140 - 48)/10 = 92/10 ≈ 9.2 units
Part B: Equilibrium Price, Output, and Profits
Equilibrium total quantity:
Q ≈ 9.6 + 9.2 = 18.8 units
Market price:
P = 145 - 5(18.8) = 145 - 94 = $51
Firm 1 profit:
π₁ = (51 - 3) 9.6 ≈ 48 9.6 ≈ $460.8
Firm 2 profit:
π₂ = (51 - 5) 9.2 ≈ 46 9.2 ≈ $423.2
Part C: Stackelberg Leadership with Firm 1 as Leader
In the Stackelberg model, Firm 1 chooses Q₁ first, anticipating Firm 2’s reaction Q₂ = (140 - 5 Q₁)/10. Revenue for Firm 1:
π₁ = (145 - 5(Q₁ + Q₂)) Q₁ - 3 Q₁
Substitute Q₂:
π₁ = (145 - 5 Q₁ - 5 * (140 - 5 Q₁)/10) Q₁ - 3 Q₁
Calculate inside parentheses:
145 - 5 Q₁ - ( (5 140) - (5 5 Q₁) ) / 10 = 145 - 5 Q₁ - (700 - 25 Q₁)/10
Express numerator:
(145 * 10) - (700 - 25 Q₁) = 1450 - 700 + 25 Q₁ = 750 + 25 Q₁
Thus, revenue becomes:
π₁ = ( (750 + 25 Q₁) / 10 ) * Q₁ - 3 Q₁ = (75 + 2.5 Q₁) Q₁ - 3 Q₁
Profit function:
π₁ = 75 Q₁ + 2.5 Q₁² - 3 Q₁ = (75 - 3) Q₁ + 2.5 Q₁² = 72 Q₁ + 2.5 Q₁²
Maximize with respect to Q₁:
dπ₁/dQ₁ = 72 + 5 Q₁ = 0
Q₁ = -72 / 5 ≈ -14.4 units, which is infeasible (negative output).
This suggests that with the given parameters, the leader’s optimal output is zero, which then results in the follower producing Q₂ based on Q₁=0, leading to the previously calculated equilibrium. Alternatively, the market conditions imply that only the lower output levels are sustainable under these parameters, reflecting strategic limitations or the importance of the costs and demand parameters in the model.
Conclusion
The analysis