Sure, I'd be happy to help you! Let's start with the Bertrand model.
In the Bertrand model, firms compete in terms of price. To find the equilibrium price and total output, we need to determine the point where the two firms' prices and quantities are equal.
a) Bertrand Model:
1. Start by setting the prices of both firms equal to their marginal costs (P = MC).
- Firm A's marginal cost is the derivative of its total cost function with respect to output (Q):
MC_A = d(TC_A)/dQ = d(0.05Q^2 + 100)/dQ = 0.1Q
- Firm B's marginal cost is also 0.1Q since they have the same cost function.
2. Set P = MC for both firms:
- P_A = MC_A = 0.1Q
- P_B = MC_B = 0.1Q
3. Set the prices of both firms equal to each other and solve for Q:
0.1Q_A = 0.1Q_B
4. Solve for Q, the total output, in terms of P (using the market demand function Q = 1000 - 5P):
0.1Q_A = 0.1Q_B = Q
0.1Q = 1000 - 5P
Q = (1000 - 5P)/0.1
5. Substitute Q back into one of the firms' price equations to find the equilibrium price:
P_A = 0.1Q = 0.1((1000 - 5P)/0.1) = 1000 - 5P
6. Solve for P to find the equilibrium price:
5P + 0.1P = 1000
5.1P = 1000
P = 196.08 (rounded to two decimal places)
7. Substitute P back into the Q equation to find the equilibrium total output:
Q = (1000 - 5P)/0.1 = (1000 - 5(196.08))/0.1 = 803.92 (rounded to two decimal places)
Therefore, in the Bertrand model, the equilibrium price is approximately $196.08, and the total output is approximately 803.92.
I hope this helps! Let me know if you have any more questions or if you'd like me to explain the Cournot model as well.