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I have this question:

Your business is the dominant firm but there exists a competitive fringe.
The competitive fringe produces with total cost: $\; c_𝑓(q_f) = 3π‘ž_f$. There have also been some changes to the costs of your firm. The costs for your firm are now: $\; c_d(q_d) = 2π‘ž_𝑑$ and market demand is now: $\; 𝑝 = 10 βˆ’ 𝑄$ where, $Q=q_f+q_d$. Finally, note that the change in total costs for your firm has also produced a capacity constraint. Your firm cannot expand beyond $π‘ž_𝑑 ≀ 7$.

a) Given these changes to the market that you supply, compute your firm’s output and profits.

is this correct?
Market demand: $p = 10 - Q$
Fringe supply: $MC_f = 3$

Equating demand and supply: $10 - Q = 3 \implies Q=7$

$Q_d = 5$

Total cost (TC) for your firm: $C=c_d = 2q_d = 2 \cdot 5 = 10$

Revenue for your firm: $R = p \cdot q_d =(10-q_d)q_d=(10-5)=25$

Profits for your firm: $\pi_d = R - c_d=25-10=15$

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The question is a bit unclear to me, but I am guessing this is what we need to do.

We can think of the situation as a dynamic game where the competitive firm takes the price as given and makes a supply decision then the dominant firm decides how much to supply and what price to set. We are given the following information:

Market demand$: \quad p=10-Q$ where, $Q=q_f+q_d$
Cost functions$: \quad c_d(q_d)=2q_d \; , \; c_f(q_f)=3q_f$

Let us denote the dominant firm as $d$ and the competitive firms as $f$
$\bullet$ Set of Players: $\{d,f\}$
$\bullet$ Action sets of players: $A_d=\{(p,q_d)\in[0,10]\times [0,7]\}$ and $A_f=\{q_f\geq 0\}$
$\bullet$ payoffs: $\pi_i:A_d \times A_f \to \mathbb{R} $ where $i \in \{d,f\}$

Since this is a dynamic game we can solve the above using backward induction

Stage 1: Competitive firm's problem $$\begin{align} &\max_{q_f\geq0} \quad \pi_f=pq_f-3q_f \\ \text{gives: } & \quad q_f^*(p)\in\begin{cases}\phi &\text{if }p>3 \\ \mathbb{R}_+ & \text{if }p=3 \\ \{0\} & \text{if }p<3\end{cases} \end{align}$$

Stage 2: Dominant firm's problem
$$\begin{align} \max_{0 \leq p \leq 10\\ 0 \leq q_d \leq 7} \quad & \pi_d=pq_d -2q_d \\ \textrm{s.t. } \quad & p=10-(q_f+q_d) \\ & q_f(p)\in\begin{cases}\phi &\text{if }p>3 \\ \mathbb{R}_+ & \text{if }p=3 \\ \{0\} & \text{if }p<3\end{cases} \\ \\ \end{align}$$

For $p>3:$ There is excess supply as the competitive firm would want to supply as many units as he can for $p>3$. Due to excess supply, we cannot have $p>3$ at equilibrium.

For $p<3:$ Market demand exceeds 7 units i.e., $Q>7$ but market supply is at most 7 because the competitive firm supplies 0 units for $p<3$ and the dominant firm can supply at most 7 units because of the capacity constraint, in symbols $Q>7\geq q_f+q_d$. Because of excess demand, there is no equilibrium for the above game where $p<3$

For $p=3:$ Market demand is $Q=7$ and the market supply can be obtained by solving stage 2 for $p=3$ $$\begin{align} \therefore \quad \max_{0 \leq q_d \leq 7} \quad &\pi_d=3q_d-2q_d=q_d \\ \textrm{s.t. } \quad & 3=10-q_f-q_d,\\ & q_f(p=3)\in \mathbb{R}_+ \end{align}$$ Notice that the objective is increasing in own output, so the above is maximized at $q_d=7$. Here market demand equals market supply, so $p=3$ is an equilibrium price.

Therefore, $\boxed{(p^*,q_d^*,q_f^*)=(3,7,0)}$ is the subgame perfect equilibrium (SPE) of the above game. Also, at SPE $\pi_d=7$ and $\pi_f=0$

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