<p>The question is a bit unclear to me, but I am guessing this is what we need to do.</p>
<p>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:</p>
<p>Market demand<span class="math-container">$: \quad p=10-Q$</span>  where, <span class="math-container">$Q=q_f+q_d$</span><br />
Cost functions<span class="math-container">$: \quad c_d(q_d)=2q_d \; , \; c_f(q_f)=3q_f$</span></p>
<p>Let us denote the dominant firm as <span class="math-container">$d$</span> and the competitive firms as <span class="math-container">$f$</span><br />
<span class="math-container">$\bullet$</span> Set of Players: <span class="math-container">$\{d,f\}$</span><br />
<span class="math-container">$\bullet$</span> Action sets of players: <span class="math-container">$A_d=\{(p,q_d)\in[0,10]\times [0,7]\}$</span> and <span class="math-container">$A_f=\{q_f\geq 0\}$</span><br />
<span class="math-container">$\bullet$</span> payoffs: <span class="math-container">$\pi_i:A_d \times A_f \to \mathbb{R} $</span> where <span class="math-container">$i \in \{d,f\}$</span></p>
<p>Since this is a dynamic game we can solve the above using backward induction</p>
<p><em><strong>Stage 1:</strong></em> <em>Competitive firm's problem</em>
<span class="math-container">$$\begin{align}
&amp;\max_{q_f\geq0} \quad  \pi_f=pq_f-3q_f \\
\text{gives: } &amp; \quad q_f^*(p)\in\begin{cases}\phi &amp;\text{if }p&gt;3 \\ \mathbb{R}_+ &amp; \text{if }p=3 \\ \{0\} &amp; \text{if }p&lt;3\end{cases}
\end{align}$$</span></p>
<p><em><strong>Stage 2:</strong></em> <em>Dominant firm's problem</em><br />
<span class="math-container">$$\begin{align} 
\max_{0 \leq p \leq 10\\ 0 \leq q_d \leq 7} \quad &amp; \pi_d=pq_d -2q_d \\
\textrm{s.t. } \quad &amp; p=10-(q_f+q_d) \\
&amp; q_f(p)\in\begin{cases}\phi &amp;\text{if }p&gt;3 \\ \mathbb{R}_+ &amp; \text{if }p=3 \\ \{0\}  &amp; \text{if }p&lt;3\end{cases} \\ \\
\end{align}$$</span></p>
<p><em>For <span class="math-container">$p&gt;3:$</span></em> There is excess supply as the competitive firm would want to supply as many units as he can for <span class="math-container">$p&gt;3$</span>. Due to excess supply, we cannot have <span class="math-container">$p&gt;3$</span> at equilibrium.</p>
<p><em>For <span class="math-container">$p&lt;3:$</span></em> Market demand exceeds 7 units i.e., <span class="math-container">$Q&gt;7$</span> but market supply is at most 7 because the competitive firm supplies 0 units for <span class="math-container">$p&lt;3$</span> and the dominant firm can supply at most 7 units because of the capacity constraint, in symbols <span class="math-container">$Q&gt;7\geq q_f+q_d$</span>. Because of excess demand, there is no equilibrium for the above game where <span class="math-container">$p&lt;3$</span></p>
<p><em>For <span class="math-container">$p=3:$</span></em> Market demand is <span class="math-container">$Q=7$</span> and the market supply can be obtained by solving stage 2 for <span class="math-container">$p=3$</span>
<span class="math-container">$$\begin{align} 
\therefore \quad \max_{0 \leq q_d \leq 7} \quad &amp;\pi_d=3q_d-2q_d=q_d \\
\textrm{s.t. } \quad &amp; 3=10-q_f-q_d,\\ &amp; q_f(p=3)\in \mathbb{R}_+
\end{align}$$</span>
Notice that the objective is increasing in own output, so the above is maximized at <span class="math-container">$q_d=7$</span>. Here market demand equals market supply, so <span class="math-container">$p=3$</span> is an equilibrium price.</p>
<p>Therefore, <span class="math-container">$\boxed{(p^*,q_d^*,q_f^*)=(3,7,0)}$</span> is the subgame perfect equilibrium (SPE) of the above game. Also, at SPE <span class="math-container">$\pi_d=7$</span> and <span class="math-container">$\pi_f=0$</span></p>

CC BY-SA 4.0
2023-06-04T12:12:19.583Z
2023-06-04T12:12:19.583Z
mynameparv
<p>I have this question:</p>
<blockquote>
<p>Your business is the dominant firm but there exists a competitive fringe.<br />
The competitive fringe produces with total cost:  <span class="math-container">$\; c_(q_f) = 3_f$</span>.
There have also been some changes to the costs of your firm.  The costs for your firm are now: <span class="math-container">$\; c_d(q_d) = 2_$</span> and market demand is now:
<span class="math-container">$\;  = 10 − $</span> where, <span class="math-container">$Q=q_f+q_d$</span>.  Finally, note that the change in total costs for your firm has also produced a capacity constraint.  Your firm cannot expand beyond <span class="math-container">$_ ≤ 7$</span>.</p>
<p><em>a)</em>   Given these changes to the market that you supply, compute your firm’s output and profits.</p>
</blockquote>
<p>is this correct?<br />
Market demand: <span class="math-container">$p = 10 - Q$</span><br />
Fringe supply: <span class="math-container">$MC_f = 3$</span></p>
<p>Equating demand and supply:
<span class="math-container">$10 - Q = 3 \implies Q=7$</span></p>
<p><span class="math-container">$Q_d = 5$</span></p>
<p>Total cost (TC) for your firm: <span class="math-container">$C=c_d = 2q_d = 2 \cdot 5 = 10$</span></p>
<p>Revenue for your firm: <span class="math-container">$R = p \cdot q_d =(10-q_d)q_d=(10-5)=25$</span></p>
<p>Profits for your firm: <span class="math-container">$\pi_d = R - c_d=25-10=15$</span></p>

<microeconomics><game-theory><markets><profit-maximization><dynamic-games>
Profit Maximizing with a competitive fringe
John Phillip
# Profit Maximizing with a competitive fringe

(Asked by: username_0 on 2023-06-04)

I have this question:

> Your business is the dominant firm but there exists a competitive fringe.  
> The competitive fringe produces with total cost: $\; c\_(q\_f) = 3\_f$.
> There have also been some changes to the costs of your firm. The costs for your firm are now: $\; c\_d(q\_d) = 2\_$ and market demand is now:
> $\; = 10 − $ 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 $\_ ≤ 7$.
>
> *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: $\pi\_d = R - c\_d=25-10=15$

**Tags:** microeconomics, game-theory, markets, profit-maximization, dynamic-games

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## 1 answers

### Answer by username_1 on 2023-06-04. Score: 3

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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