Question

Consider the following market for hooded garments in which two firms, Oliver Queen, Inc. and Bruce Wayne, LLC are competing. They make equally good cowls. Suppose the inverse-demand for cowls is P - 280 - Q. Each firm has to choose its quantity produced at the same time without communicating. Further, suppose that both Oliver Queen, Inc. and Bruce Wayne, LLC can make hoods at the same marginal cost of 40. What is the Nash Equilibrium of this game? What are the profits of each firm? (10 points)

Answer 1

Profit maximization problem for firm one:
$\dpi{80} \fn_phv \small max \ (280-q_1-q_2)q_1-40q_1$
Differentiating the objective with respect to the quantity and setting the derivative equal to zero:
$\dpi{80} \fn_phv \small 280-2q_1-q_2-40=0\rightarrow q_1=\frac{240-q_2}{2}$
Best response function for firm one is:
$\dpi{80} \fn_phv \small BR_1(q_2)=\begin{cases}\frac{240-q_2}{2}&q_2\leq240\\0 & otherwise \end{cases}$
By symmetry, best response function of firm two is:
$\dpi{80} \fn_phv \small BR_2(q_1)=\begin{cases}\frac{240-q_1}{2}&q_1\leq240\\0 & otherwise \end{cases}$
Solving the two functions:
$\dpi{80} \fn_phv \small q_2=\frac{240-(\frac{240-q_2}{2})}{2}\rightarrow q_2=80,q_1=80$
At these quantities, market price is 120.
Profit earned by each firm is:
$\dpi{80} \fn_phv \small \pi_1=\pi_2=120*80-40*80=6400$