Question

The inverse market demand in a homogeneous product Cournot duopoly is P=100-2(Q1+Q2), and the costs are given by C(Q1) = 12Q1 and C(Q2) = 20Q2. The implied marginal costs are $12 for firm 1 and $20 for firm 2.
a. Determine the reaction function for firm 1.
b. Determine the reaction function for firm 2.
c. Calculate the Cournot equilibrium price and quantity.
d. Suppose firm 1 is a monopoly (firm 2 does not exist), what is firm 1's monopoly output and price?
e. How does the monopoly price and quantity comparing with Cournot equilibrium in part (c)?

Answer 1

Answer:

Following are the solution to the given points:

Explanation:

$Q_1= 22-\frac{Q_2}{2}\\\\Q_2= 20-\frac{Q_1}{2}$

Replacing the $Q_1$ value in the equation mentioned above:

$Q_2= 20- (\frac{(22-\frac{Q_2}{2})}{2})\\\\Q_2= 20-11+\frac{Q_2}{4}\\\\3\frac{Q_2}{4}=9\\\\Q_2= \frac{(9\times4)}{3}\\\\Q_2= 12$

Substituting the value of $Q_2$:

$Q_1= 22- \frac{Q_2}{2}\\\\Q_1= 22-6\\\\Q_1=16$

Now

$P= 100-2(Q_1+Q_2)\\\\P= 100-2(16+12)\\\\P=100- 56\\\\P= 44$

In this firm a monopoly: (Q2 is 0 now)

Profit maximization is at MR=MC

as per the equation$P= 100-2Q_1$

$P.Q1= 100Q_1-2Q_1^{2}\\\\MR= 100-4Q_1\\\\MC= 12$

$100-4Q_1=12\\\\4Q_1=88\\\\Q_1= 22\\\\and \ P= 100-2(22)= 56.$