# bertrand duopoly with discrete price

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.

• 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. Commented Apr 10 at 17:56
• My bad. I did not read the last part. Commented Apr 10 at 18:54

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.

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.