Return to the example from lessons 2.2 and 2.3: Firm 1 and Firm 2 have a constant \(MC=AC=8\). The market (inverse) demand curve is given by:

\[\begin{aligned} P&=200-2Q\\ Q&=q_1+q_2\\ \end{aligned}\]

- Suppose Firm 1 is the Leader and Firm 2 is the follower. Find the Stackelberg Nash Equilibrium quantity for each firm. Hint, the Cournot reaction functions you found before were:

\[\begin{aligned} q_1^\star & = 48 - 0.5 * q_2\\ q_2^\star & = 48 - 0.5 * q_1\\ \end{aligned}\]

Substitute follower’s reaction function into market (inverse) demand function

\[\begin{align*} P&=200-2q_{1}-2q_2 && \text{The inverse market demand function}\\ P&=200-2q_{1}-2(48-0.5q_{1}) && \text{Plugging in Firm 2's reaction function for} q_2\\ P&=200-2q_{1}-96+1q_{1} && \text{Multiplying by }-3\\ P&=104-q_{1} && \text{Simplifying the right}\\ \end{align*}\]

- Find \(MR\) for Firm 1 from market demand

\[MR_{1}=104-2q_{1}\]

\[\begin{align*} MR&=MC && \text{Profit-max condition}\\ 104-2q_{1}&=8 && \text{Plugging in}\\ 104&=8+2q_{1} && \text{Adding }2q_{1} \text{ to both sides}\\ 96&=2q_{1} && \text{Subtracting 20 from both sides}\\ 48&=q_{1}^* && \text{Dividing both sides by 2} \\ \end{align*}\]

Firm 2 will respond:

\[\begin{align*} q_2^*&=48-0.5q_{1}\\ q_2^*&=48-0.5(48)\\ q_2^*&=48-24\\ q_2^*&=24\\ \end{align*}\]

- Find the market price.

With \(q^*_{1}=48\) and \(q^*_2=24\), this sets a market price of

\[\begin{align*} P&=200-2Q\\ P&=200-2(72)\\ P&=56\\ \end{align*}\]

- Find the profit for each firm. Compare their profits under Stackelberg competition to their profits under Cournot competition (from lesson 2.2).

Profit for Firm 1 is

\[\begin{align*} \pi_{1}&=q_{1}(P-c)\\ \pi_{1}&=48(56-8)\\ \pi_{1}&=\$2,304\\ \end{align*}\]

Profit for Firm 2 is

\[\begin{align*} \pi_{2}&=q_{2}(P-c)\\ \pi_{2}&=24(56-8)\\ \pi_{2}&=\$1,152\\ \end{align*}\]