Answer :
To solve this problem, we need to find the equilibrium quantity, price, and profits for both Coke and Pepsi when they are competing in the soda market by selecting their output levels, [tex]Q_C[/tex] for Coke and [tex]Q_P[/tex] for Pepsi.
Step 1: Understanding the Demand Curve
The demand curve provided is:
[tex]P = 200 - Q_C - Q_P[/tex]
Where:
- [tex]P[/tex] is the price of soda.
Step 2: Marginal Cost
The average (and marginal) cost of production for both companies is given as 20.
Step 3: Finding the Reaction Functions
In a Cournot competition (where firms choose quantities simultaneously to maximize profits), each firm's output depends on the output choice of the other firm. We'll derive the reaction functions for both companies.
For Coke:
Total Revenue for Coke: [tex]TR_C = P \times Q_C = (200 - Q_C - Q_P) \times Q_C[/tex]
Profit for Coke: [tex]\pi_C = TR_C - TC_C = (200 - Q_C - Q_P) \times Q_C - 20 \times Q_C[/tex]
To find Coke's output that maximizes its profit, we differentiate the profit function with respect to [tex]Q_C[/tex] and set it to zero:
[tex]\frac{d\pi_C}{dQ_C} = 200 - 2Q_C - Q_P - 20 = 0[/tex]
Solve for [tex]Q_C[/tex]:
[tex]Q_C = 90 - \frac{Q_P}{2}[/tex]
For Pepsi:
Total Revenue for Pepsi: [tex]TR_P = P \times Q_P = (200 - Q_C - Q_P) \times Q_P[/tex]
Profit for Pepsi: [tex]\pi_P = TR_P - TC_P = (200 - Q_C - Q_P) \times Q_P - 20 \times Q_P[/tex]
To find Pepsi's output that maximizes its profit, differentiate the profit function with respect to [tex]Q_P[/tex] and set it to zero:
[tex]\frac{d\pi_P}{dQ_P} = 200 - Q_C - 2Q_P - 20 = 0[/tex]
Solve for [tex]Q_P[/tex]:
[tex]Q_P = 90 - \frac{Q_C}{2}[/tex]
Step 4: Solving the Reaction Functions Simultaneously
Substitute [tex]Q_P = 90 - \frac{Q_C}{2}[/tex] in Coke's reaction function:
[tex]Q_C = 90 - \frac{1}{2}(90 - \frac{Q_C}{2})[/tex]
Solve for [tex]Q_C[/tex]:
[tex]Q_C = 60[/tex]
Now, substitute [tex]Q_C = 60[/tex] in Pepsi's reaction function:
[tex]Q_P = 90 - \frac{60}{2} = 60[/tex]
Step 5: Find the Equilibrium Price
Substitute [tex]Q_C = 60[/tex] and [tex]Q_P = 60[/tex] into the demand equation to find the price:
[tex]P = 200 - 60 - 60 = 80[/tex]
Step 6: Calculate Profits
For Coke:
[tex]\pi_C = (P - MC) \times Q_C = (80 - 20) \times 60 = 3600[/tex]
For Pepsi:
[tex]\pi_P = (P - MC) \times Q_P = (80 - 20) \times 60 = 3600[/tex]
Conclusion:
- Equilibrium Quantities: Both Coke and Pepsi produce 60 units.
- Equilibrium Price: The price is 80.
- Profits: Each firm earns a profit of 3600.
