Suppose there are only two producers, A and B, in a market producing the same good with the same total cost function
TC=0.05 Q^2 +100.
The market demand is given by
Q=1000-5P.
a) Find the equilibrium price and total output in the Bertrand model.
b) Find the equilibrium price and total output in the Cournot model.
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Can someone please teach me how to compute these? If the marginal cost were constant, then I think I know how to answer these; for this problem, it's the cost function that is confusing me.
For the Bertrand model, we set P=MC, and solve for the output, but what is the MC in this case? There are two firms, so do we have to add them to get the MC? But how to add them? What is the equation that I have to solve?
I hope someone can be kind enough to help me out! Thank you so much!
MR for the INDUSTRY is:
MR=200-(2/5)Q
The MC for EACH FIRM is MC=0.1q.
The trouble is that the two equations have different context, the MR is for the whole industry, but the MC is for each firm.
I am not sure how to bring this to a common footing. How can we add the MC?
What is the equation that we have to solve to get the Bertrand equilibrium price and total output in this case?
I hope someone can help! Thanks a lot!
But I think the Bertrand equilibrium says P=MC (rather than MR=MC). It's a price strategy.
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.