Part 1 Suppose the inverse market demand function for a two-firm Cournot model is given by P=49-Q where Q=q_1+q_2 is the total output produced in the market, P is the market price, q_1 is the quantity of output produced by firm 1, q_2 is the quantity of output produced by firm 2. The marginal costs of the two firms are MC1=2 and MC2=3. For a linear inverse demand function P=a+bq, the marginal revenue is given by MR=a+2bq.
1. Find the marginal revenue MR1 of firm 1 as a function of q_1 and q_2.
2. Setting MR1=MC1, determine the reaction function of firm 1.
3. Find the marginal revenue MR2 of firm 2 as a function of q_1 and q_2.
4. Setting MR2=MC2, determine the reaction function of firm 2.
5. What quantity of output will each firm produce? That is, use the two reactions functions to find q_1 and q_2.
6. What is the market price P.
7. Determine the profits of each firm assuming there are no fixed costs.
Part 2 Now, suppose the inverse market demand function for a two-firm Stackelberg model is given by P=49-Q where Q=q_1+q_2 is the total output produced in the market, P is the market price, q_1 is the quantity of output produced by firm 1, q_2 is the quantity of output produced by firm 2. Firm is leader and firm 2 is follower. The marginal costs of the two firms are MC1=2 and MC2=3.
1. What is the follower’s reaction function?
2. Determine the equilibrium output level for both the leader and the follower.
3. Determine the equilibrium market price.
4. Determine the profits of the leader and the follower assuming there are no fixed costs.
Part 3 Use information from parts 1 & 2 to compare the output levels and profits in settings characterized by Cournot and Stackelberg.
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