<p>By &quot;discrete price&quot;, I assume both firms can charge price from the set of non-negative integers.</p>
<ul>
<li>Define the players as <span class="math-container">$N = \{A,B\}$</span></li>
<li>Price competition</li>
<li>Action set of both players = <span class="math-container">$\mathbb{Z_+}$</span></li>
<li>Assuming simultaneous move game and perfect information to both players</li>
<li>Quantity demanded is : <span class="math-container">$Q^d(P) =120 -P $</span></li>
<li><span class="math-container">$C_A(Q_A) = 10Q_A $</span></li>
<li><span class="math-container">$C_B(Q_B) = 21Q_B$</span></li>
</ul>
<p>Since both firms compete in prices, the firms that set lower prices gets quantity demanded as per the demand function and if prices are equal, I assume both firms share the quantity demanded equally at that price. Profits in this Bertrand duopoly can be defined as:-</p>
<p><span class="math-container">$\pi_i(P_i,P_j)$</span> = <span class="math-container">$\left\{\begin{matrix}
0&amp; \text{if}\: P_i &gt;P_j \\ 
P_i .\frac{Q^d(P_i)}{2}- C_i(\frac{Q^d(P_i)}{2})&amp; \text{if}\: P_i =P_j  \\
P_i {Q^d(P_i)}- C_i(Q^d(P_i))&amp; \text{if}\: P_i &lt;P_j 
\end{matrix}\right.$</span></p>
<p>where <span class="math-container">$i,j \in \{A,B\} \: \text{but} \: i\neq j$</span></p>
<p>This will yield:-</p>
<p><span class="math-container">$\pi_A(P_A,P_B)$</span> = <span class="math-container">$\left\{\begin{matrix}
0&amp; \text{if}\: P_A &gt;P_B \\ 
\frac{(P_A -10)(120-P_A)}{2}&amp; \text{if}\: P_A =P_B  \\
(P_A -10)(120-P_A)  &amp; \text{if}\: P_A &lt;P_B 
\end{matrix}\right.$</span></p>
<p>and</p>
<p><span class="math-container">$\pi_B(P_A,P_B)$</span> = <span class="math-container">$\left\{\begin{matrix}
0&amp; \text{if}\: P_B &gt;P_A \\ 
\frac{(P_B -21)(120-P_B)}{2}&amp; \text{if}\: P_B =P_A  \\
(P_A -21)(120-P_B)  &amp; \text{if}\: P_B &lt;P_A 
\end{matrix}\right.$</span></p>
<p>Best Response <span class="math-container">$BR_i(P_j) $</span> of <span class="math-container">$i^{th}$</span> firm can be found by solving following optimisation problem :-</p>
<p><span class="math-container">$ \underset{P_i \in \mathbb{Z_+}}{max} \: \pi_i(P_i,P_j)$</span></p>
<p>This yields (on solving for <span class="math-container">$i \in \{A,B\}$</span>):-</p>
<p><span class="math-container">$BR_A(P_B)$</span> = <span class="math-container">$\left\{\begin{matrix}
\{65\}&amp; \text{if} \:  P_B &gt;65 \\ 
\{P_B-1\}  &amp; \text{if} \: P_B \in \{12,13,14...,65\}  \\
\{11 \}  &amp; \text{if} \: P_B =11   \\
\{a \in \mathbb{Z_+}| a \geq 10 \}  &amp; \text{if} \: P_B =10  \\
\{ a \in \mathbb{Z_+}| a &gt; P_B \} &amp; \text{if} \: P_B &lt;10 
\end{matrix}\right.$</span></p>
<p><span class="math-container">$BR_B(P_A)$</span> = <span class="math-container">$\left\{\begin{matrix}
\{70,71\}&amp; \text{if} \: P_A &gt; 71  \\ 
\{70\}  &amp; \text{if} \: P_A =71  \\
\{P_A -1 \}  &amp; \text{if} \: P_A \in \{23,24...,70\}  \\
\{22 \}  &amp; \text{if} \: P_A =22  \\
\{x \in \mathbb{Z_+}| x \geq 21 \}  &amp; \text{if} \: P_A =21  \\
\{ x \in \mathbb{Z_+}| x &gt; P_A  \} &amp; \text{if} P_A \leq 20\: 
\end{matrix}\right.$</span></p>
<p>Nash equilibrium is all those <span class="math-container">$(P_A^*,P_B^*) \in \{(P_A,P_B) \in \mathbb{Z_+^2}| P_A \in BR_A(P_B) \wedge P_B \in BR_B(P_A) \} $</span></p>
<p>Verify that the set of nash equilibrium prices is <span class="math-container">$\{ (21,22),(20,21) ,(19,20),(18,19),(17,18),(16,17),(15,16),(14,15),(13,14),(12,13),(11,12) \}$</span></p>
<p>Notice that in each Nash equilibrium, firm B sets a higher price and hence its profit is 0 in any Nash equilibria. Maximum profit for firm A (among the nash equilibria) is when <span class="math-container">$P_A^* = 21$</span> and <span class="math-container">$P_B^* = 22$</span> which is equal to 1089.</p>
<p>Had it been the case that the firm 2 was not present, the monopoly A firm would have charged <span class="math-container">$P_A = 65 \: \text{with} \: \pi_A = 3025$</span>. This means the maximum firm A would pay to firm B to exit the market is = <span class="math-container">$3025-1089 = 1936$</span>. Notice, that firm B in Nash equilibrium made 0 profit, so minimum it needs to exit the market is 0.</p>

CC BY-SA 4.0
2024-04-11T02:41:26.143Z
2024-04-11T02:41:26.143Z
Divyam Verma
<p>Since the Marginal cost of firm 1 is <span class="math-container">$10$</span>, i.e. <span class="math-container">$\frac{\partial C_A(q)}{\partial q}=10$</span> and the Marginal cost of firm B is 21, i.e. <span class="math-container">$\frac{\partial C_B(q)}{\partial q}=21$</span> and both firms are only allowed to Charge integer prices, so firm A will charge <span class="math-container">$P=21-1=20$</span> and will enjoy entire market to itself since firm B cannot produce at this price and make losses, from the demand function we can see that at <span class="math-container">$P=20, q=100$</span>. Profits of Firm A at <span class="math-container">$P=20$</span> are <span class="math-container">$\pi^A=(20-10)100=1000$</span></p>
<p>Now if firm B is not in the market then Firm A can serve the market as a monopolist so firm A will produce that quantity at which its Marginal Revenue <span class="math-container">$MR=120-2q$</span> will be equal to it's marginal cost <span class="math-container">$MC=10$</span> to maximize it's profits, equating the two we get that  <span class="math-container">$q=55,P=65$</span> and the profits of Monopolist will be <span class="math-container">$\pi ^m=(65-10)55=3025$</span>. Notice that <span class="math-container">$\pi ^m&gt; \pi ^A$</span> Therefore the maximum amount firm A would be willing to pay to the firm B to exit the market is <span class="math-container">$\pi ^m- \pi ^A=3025-1000=2025$</span>.</p>
<p>Now maybe you can try to solve part (c) on your own.</p>

CC BY-SA 4.0
2024-04-10T16:43:44.283Z
2024-04-10T17:24:08.567Z
2024-04-10T17:24:08.567Z
econhead
<p>There are 2 firms, engaging in price competition. Total costs are <span class="math-container">$C_A(q)=10q$</span> and <span class="math-container">$C_B(q)=21q$</span> and the inverse demand is <span class="math-container">$P(q)=120-q$</span>.</p>
<ul>
<li>What is the equilibrium profits?</li>
<li>What is the maximum amount that firm <span class="math-container">$A$</span> would be willing to pay to firm <span class="math-container">$B$</span> to exit the market and the minimum amount that firm <span class="math-container">$B$</span> would accept to exit the market?</li>
</ul>
<p>So the marginal costs are <span class="math-container">$c_A=10$</span> and <span class="math-container">$c_B=21$</span>. I assume that the price of firm <span class="math-container">$A$</span> is <span class="math-container">$21-1=20$</span>. Then the <span class="math-container">$P(q)= 120 - q =20 \Rightarrow q = 100$</span>.</p>
<p>And the profit of firm <span class="math-container">$A$</span> is <span class="math-container">$(p-c)q = (20-10)\times 100 = 1000$</span>.
Is this the right approach to solve this question?</p>

CC BY-SA 4.0
2024-04-10T17:56:26.360Z
Hey man you wrote under your other question that you will read the [meta post](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions), but you still seem to be violating most of its points? Your Q contains specific numbers and is a scan of a question. Also, if you want to improve it instead of reposting you can edit the earlier version.
Giskard
CC BY-SA 4.0
2024-04-10T18:54:05.087Z
My bad. I did not read the last part.
Derin dur
<microeconomics><game-theory><industrial-organisation>
bertrand duopoly with discrete price
Derin dur
# bertrand duopoly with discrete price

(Asked by: username_0 on 2024-04-10)

There are 2 firms, engaging in price competition. Total costs are $C\_A(q)=10q$ and $C\_B(q)=21q$ and the inverse demand is $P(q)=120-q$.

* What is the equilibrium profits?
* What is the maximum amount that firm $A$ would be willing to pay to firm $B$ to exit the market and the minimum amount that firm $B$ would accept to exit the market?

So the marginal costs are $c\_A=10$ and $c\_B=21$. I assume that the price of firm $A$ is $21-1=20$. Then the $P(q)= 120 - q =20 \Rightarrow q = 100$.

And the profit of firm $A$ is $(p-c)q = (20-10)\times 100 = 1000$.
Is this the right approach to solve this question?

**Tags:** microeconomics, game-theory, industrial-organisation

> **Comments:**
> **username_1**: Hey man you wrote under your other question that you will read the [meta post](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions), but you still seem to be violating most of its points? Your Q contains specific numbers and is a scan of a question. Also, if you want to improve it instead of reposting you can edit the earlier version.  
> **username_0**: My bad. I did not read the last part.  

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## 2 answers

### Answer by username_2 on 2024-04-11. Score: 2

By "discrete price", I assume both firms can charge price from the set of non-negative integers.

* Define the players as $N = \{A,B\}$
* Price competition
* Action set of both players = $\mathbb{Z\_+}$
* Assuming simultaneous move game and perfect information to both players
* Quantity demanded is : $Q^d(P) =120 -P $
* $C\_A(Q\_A) = 10Q\_A $
* $C\_B(Q\_B) = 21Q\_B$

Since both firms compete in prices, the firms that set lower prices gets quantity demanded as per the demand function and if prices are equal, I assume both firms share the quantity demanded equally at that price. Profits in this Bertrand duopoly can be defined as:-

$\pi\_i(P\_i,P\_j)$ = $\left\{\begin{matrix}
0& \text{if}\: P\_i >P\_j \\
P\_i .\frac{Q^d(P\_i)}{2}- C\_i(\frac{Q^d(P\_i)}{2})& \text{if}\: P\_i =P\_j \\
P\_i {Q^d(P\_i)}- C\_i(Q^d(P\_i))& \text{if}\: P\_i

where $i,j \in \{A,B\} \: \text{but} \: i\neq j$

This will yield:-

$\pi\_A(P\_A,P\_B)$ = $\left\{\begin{matrix}
0& \text{if}\: P\_A >P\_B \\
\frac{(P\_A -10)(120-P\_A)}{2}& \text{if}\: P\_A =P\_B \\
(P\_A -10)(120-P\_A) & \text{if}\: P\_A

and

$\pi\_B(P\_A,P\_B)$ = $\left\{\begin{matrix}
0& \text{if}\: P\_B >P\_A \\
\frac{(P\_B -21)(120-P\_B)}{2}& \text{if}\: P\_B =P\_A \\
(P\_A -21)(120-P\_B) & \text{if}\: P\_B

Best Response $BR\_i(P\_j) $ of $i^{th}$ firm can be found by solving following optimisation problem :-

$ \underset{P\_i \in \mathbb{Z\_+}}{max} \: \pi\_i(P\_i,P\_j)$

This yields (on solving for $i \in \{A,B\}$):-

$BR\_A(P\_B)$ = $\left\{\begin{matrix}
\{65\}& \text{if} \: P\_B >65 \\
\{P\_B-1\} & \text{if} \: P\_B \in \{12,13,14...,65\} \\
\{11 \} & \text{if} \: P\_B =11 \\
\{a \in \mathbb{Z\_+}| a \geq 10 \} & \text{if} \: P\_B =10 \\
\{ a \in \mathbb{Z\_+}| a > P\_B \} & \text{if} \: P\_B <10
\end{matrix}\right.$

$BR\_B(P\_A)$ = $\left\{\begin{matrix}
\{70,71\}& \text{if} \: P\_A > 71 \\
\{70\} & \text{if} \: P\_A =71 \\
\{P\_A -1 \} & \text{if} \: P\_A \in \{23,24...,70\} \\
\{22 \} & \text{if} \: P\_A =22 \\
\{x \in \mathbb{Z\_+}| x \geq 21 \} & \text{if} \: P\_A =21 \\
\{ x \in \mathbb{Z\_+}| x > P\_A \} & \text{if} P\_A \leq 20\:
\end{matrix}\right.$

Nash equilibrium is all those $(P\_A^\*,P\_B^\*) \in \{(P\_A,P\_B) \in \mathbb{Z\_+^2}| P\_A \in BR\_A(P\_B) \wedge P\_B \in BR\_B(P\_A) \} $

Verify that the set of nash equilibrium prices is $\{ (21,22),(20,21) ,(19,20),(18,19),(17,18),(16,17),(15,16),(14,15),(13,14),(12,13),(11,12) \}$

Notice that in each Nash equilibrium, firm B sets a higher price and hence its profit is 0 in any Nash equilibria. Maximum profit for firm A (among the nash equilibria) is when $P\_A^\* = 21$ and $P\_B^\* = 22$ which is equal to 1089.

Had it been the case that the firm 2 was not present, the monopoly A firm would have charged $P\_A = 65 \: \text{with} \: \pi\_A = 3025$. This means the maximum firm A would pay to firm B to exit the market is = $3025-1089 = 1936$. Notice, that firm B in Nash equilibrium made 0 profit, so minimum it needs to exit the market is 0.


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### Answer by username_3 on 2024-04-10. Score: 0

Since the Marginal cost of firm 1 is $10$, i.e. $\frac{\partial C\_A(q)}{\partial q}=10$ and the Marginal cost of firm B is 21, i.e. $\frac{\partial C\_B(q)}{\partial q}=21$ and both firms are only allowed to Charge integer prices, so firm A will charge $P=21-1=20$ and will enjoy entire market to itself since firm B cannot produce at this price and make losses, from the demand function we can see that at $P=20, q=100$. Profits of Firm A at $P=20$ are $\pi^A=(20-10)100=1000$

Now if firm B is not in the market then Firm A can serve the market as a monopolist so firm A will produce that quantity at which its Marginal Revenue $MR=120-2q$ will be equal to it's marginal cost $MC=10$ to maximize it's profits, equating the two we get that $q=55,P=65$ and the profits of Monopolist will be $\pi ^m=(65-10)55=3025$. Notice that $\pi ^m> \pi ^A$ Therefore the maximum amount firm A would be willing to pay to the firm B to exit the market is $\pi ^m- \pi ^A=3025-1000=2025$.

Now maybe you can try to solve part (c) on your own.


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