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There is a 2 firm, engaging in price competition. Total costs are CA(q)=10q and CB(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 to exit the market and the minium amount that firm B would accept to exit the market?

So the marginal costs are ca=10 and cb=21. I assume that the price of firm A is 21-1=20. Then the P(q)= 120 - q =20 >> q = 100. And the profit of firm A is (p-c)*q = (20-10)*100 = 1000 Is this the right approach to solve this question? Economics is not my main area of study and I tried to solve this question from what I found on the internet. If you enlighten me I will very pleased.

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  • $\begingroup$ Hey man you wrote under your other question that you will read the meta post, 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. $\endgroup$
    – Giskard
    Commented Apr 10 at 17:56
  • $\begingroup$ My bad. I did not read the last part. $\endgroup$
    – Derin dur
    Commented Apr 10 at 18:54

2 Answers 2

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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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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:-

$\pi_i(P_i,P_j)$ = $\left\{\begin{matrix} 0& \text{if}\: P_i >P_j \\ P_i .\frac{Q^d(P_i)}{2}- C_i(\frac{Q^d(P_i)}{2})& \text{if}\: P_i =P_j \\ P_i {Q^d(P_i)}- C_i(Q^d(P_i))& \text{if}\: P_i <P_j \end{matrix}\right.$

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

This will yield:-

$\pi_A(P_A,P_B)$ = $\left\{\begin{matrix} 0& \text{if}\: P_A >P_B \\ \frac{(P_A -10)(120-P_A)}{2}& \text{if}\: P_A =P_B \\ (P_A -10)(120-P_A) & \text{if}\: P_A <P_B \end{matrix}\right.$

and

$\pi_B(P_A,P_B)$ = $\left\{\begin{matrix} 0& \text{if}\: P_B >P_A \\ \frac{(P_B -21)(120-P_B)}{2}& \text{if}\: P_B =P_A \\ (P_A -21)(120-P_B) & \text{if}\: P_B <P_A \end{matrix}\right.$

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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