(Repeated Games). Suppose two firms compete in micro-chip industry. Each period firm 1 produces q1 chips and firm two produces q2 chips and the firms face a demand curve of P = 1000−20Q, where Q = q1 +q2. Both firms have a constant marginal cost of $40 per chip, C(qi) = 40qi.
- What are the static Nash equilibrium strategies for this market? What are equilibrium profits when the market operates for a single period?
- Suppose the two firms agree to maximize joint profits rather than individual profits and share the proceeds equally. How many chips does each firm agree to make? What are firms profits for a single period?
can you please help with these questions?
Ad 1.
The profit of firm 1 is $\Pi_1=P(q_1,q_2)\cdot q_1-C(q_1)=(1000-20(q_1+q_2))\cdot q_1-40q_1 $
The profit of firm 2 is $\Pi_2=P(q_1,q_2)\cdot q_2-C(q_2)=(1000-20(q_1+q_2))\cdot q_2-40q_2 $
Calculate the (partial) derivatives $\frac{\partial \Pi_2}{\partial q_1}$ and $\frac{\partial \Pi_2}{\partial q_2 }$ and set both terms equal to zero. Then calculate the optimal values of $q_1$ and $q_2$
Ad 2.
Maximize $P(Q)=(1000−20Q)\cdot Q-40Q$ w.r.t $Q$
Each firm receives the half of the maximized profit $P(Q)$.