In lecture we saw the Cournot competition model for two firms with the same cost function. Now, we are going to consider asymmetric cost functions. Assume that demand for a good is given by= ( is quantity demanded), and that there are2firms competing in quantities. Both have no fixed costs and a constant marginal cost. Firm 1 has a marginal cost1, and firm 2 has a marginal cost2. We have that>1>2.

Find the reaction functions of firms 1 and 2 in this market: how the optimal quantity produced depends on the quantity produced by the other firm.

To verify that you have found the correct reaction functions, compute the optimal1if2=100,=4,=0.01,1=2, and2=1, q1 = 50

1.) To verify that you have found the correct equilibrium, computeif=4,=0.01,1=2, 2=1, q*1 =33.33, q*2= 133.33.

What is p*?

Find the equilibrium price and the quantity produced by each firm if they compete in prices (Bertrand competition).

2.) Is p close to c1 or c2 or (c1+c2)/2 or 0 ?

3.) Is q1 close to 0 or (a-c1-c2)/b or (a-c1)/b or (a-c2)/b ?

4.) Is q2 close to (a-c1)/b or (a-c1-c2)/b or (a-c2)/b or 0 ?

Now, let's go back to the case where all firms have the same cost function. In class we saw the Cournot competition model for two firms. Now, we are going to get you through the Cournot model with three firms. Assume that demand for a good is given by=, and that there are3 firms competing in quantity with a constant marginal cost<.

Write down the maximization problem of a representative firm in the market. Solve for the reaction function of this firm: how the optimal quantity produced depends on the quantity produced by the other two firms in the market.

To verify that you have found the correct reaction function, compute the optimal1if2=40,3=60,=4,=0.01, and=2. (Note that this is not necessarily an equilibrium.)

5.) What is q1?

In the three-firm case, what will be the equilibrium price and the total quantity produced in the market? To verify that you have found the correct equilibrium, compute andif =4,=0.01, and=2.

6.) What is p?

7.) What is Q?