Suppose Firm 1 and Firm 2 have a constant \(MC=AC=8\). The market (inverse) demand curve is given by:

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

Question 1

Find the Cournot-Nash equilibrium output and profit for each firm.

Break demand apart into both firms’ output. \[\begin{align*} P&=200-2Q\\ P&=200-2(q_{1}+q_{2})\\ P&=200-2q_{1}-2q_2\\ \end{align*}\]

Solving for Firm 1, recalling that MR is twice the slope of the inverse demand curve:

\[MR_{1}=200-4q_{1}-2q_2\]

Firm 1 maximizes profit at \(q^*\) where \(MR=MC\): \[\begin{align*} MR_{1}&=MC && \text{Profit-max condition} \\ 200-4q_{1}-2q_2&=8 && \text{Plugging in}\\ 192-4q_1-2q_2&=0 && \text{Subtracting 8 from both sides}\\ 192-2q_2&=4q_1 && \text{Adding }4q_{1} \text{ to both sides}\\ 48-0.5q_1&=q_1^* && \text{Dividing both sides by 4}\\ \end{align*}\]

  • Since Firm 2 is identical, its \(q^*\) is:

\[q_2^*=48-0.5q_{1}\]

Find Nash equilibrium algebraically by plugging in one firm’s reaction curve into the other’s

\[\begin{align*} q_{1}&=48-0.5q_2 && \text{Firm 1's reaction function}\\ q_{1}&=48-0.5(48-0.5q_{1}) && \text{Plugging in Firm 2's reaction function}\\ q_{1}&=48-24+0.25q_{1} && \text{Distributing the }-0.5\\ q_{1}&=24+0.25q_{1} && \text{Subtracting}\\ 0.75q_{1}&=24 && \text{Subtracting }0.25q_{1} \text{ to both sides}\\ q_{1}&=32 && \text{Dividing by 0.75}\\ \end{align*}\]

Symmetrically, \(q_{2}=32\)

Both countries produce 32:

\[\begin{align*} P&=200-2(32)-2(32)\\ P&=\$72\\ \end{align*}\]

We can find the profit for each firm:

\[\begin{align*} \pi_{1}&=q_{1}(P-c)\\ \pi_{1}&=32(72-8)\\ \pi_{1}&=\$2,048\\ \end{align*}\]

Question 2

Find the output and profit for each firm if the two were to collude.

Suppose now both firms collude and act like a single monopolist, who sets:

\[\begin{align*} MR&=MC && \text{Profit-max condition}\\ 200-4Q&=8 && \text{Plugging in}\\ 192-4Q&=0 && \text{Subtracting 8 from both sides}\\ 192&=4Q && \text{Adding }4Q \text{ to both sides}\\ 48&=Q && \text{Dividing both sides by 4}\\ \end{align*}\]

So each firm will produce 24.

The monopoly price will then be

\[\begin{align*} P&=200-2Q\\ P&=200-2(48)\\ P&=\$104\\ \end{align*}\]

Total profit will then be:

\[\begin{align*} \Pi&=Q(P-c)\\ \Pi&=48(104-8)\\ \Pi&=\$4,608\\ \end{align*}\]

with $2,304 going to each firm

Question 3

Find the price and output if the two were to compete on price instead of quantity.

In Bertrand competition, each firm sets:

\[\begin{align*} P&=MC\\ 200-2Q&=8\\ 192-2Q&=0\\ 192&=2Q\\ 96&=Q\\ \end{align*}\]

Each firm produces 48, earning no profits.

Question 4

Suppose Firm 1 produces before Firm 2, who can see how much Firm 1 produced. Find the price, output, and profit for each firm.

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

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

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

  • Firm 2 will respond

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

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

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

  • 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

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