Two firms, called 1 and 2, compete in a Bertrand duopoly. The products are ident

Question

Two firms, called 1 and 2, compete in a Bertrand duopoly. The products are identical, so consumers buy from the firm with the lower price. Inverse demand is given by P = 9 Q. Firm 1 has a marginal cost of 1 and Firm 2 has a marginal cost of 2. Firms compete by choosing a single integer price. If the prices are identical they split demand.

(a) [10 points] If firm 2 were a monopolist what price would it set?

(b) [10 points] Can there be a Nash equilibrium where both firms set prices above 3? Why or

why not?

(c) [10 points] If both firms set prices of 3, is this a Nash equilibrium?

(d) [10 points] Is there a Nash equilibrium where one firm sets a price of 3 and the other doesn’t? Why or why not?

(e) [10 points] If both firms set prices of 2, is this a Nash equilibrium? Why or why not

Explanation / Answer

a).

Consider the given problem, here the market demand curve be, “P = 9 – Q”. Now, firm-2 be a monopolist here having MC=MC2=2.

So, if “P=9 – Q, => TR = P*Q = 9*Q – Q^2, => MR = 9 – 2*Q. So, the optimum choice of “P” and “Q” will be determined by, MR=MC condition.

=> MR = MC, => 9 – 2*Q = 2, => 2*Q = 7, => Q=7/2 = 3.5, => Q=3.5.

So, at Q=3.5, P=9-Q=9 – 3.5 =5.5.

So, if firm-2 is a monopolist then it will charge “P=5.5 and will supply Q=3.5”.

b).

Now, consider that there are 2 firms and both are producing identical product. Now, the firm-1 have MC=MC1=1 and firm-2 have MC=MC2=2. Now, let’s assume that they charging same price P=4 > 3. So, at this price total production will be “Q = 9 – 4 =5”. Now, since they are producing same product and also charging same price, => they will supply equal amount to the market. Now, since P=4 > MC1, MC2, => both the firms are earning positive profits.

Now, here each firms have an incentive to charge less “P” than it’s rival firm, to capture the entire market, to earn more profit. So, it is totally clear that both of them will think and act in a same way => both of them will reduce “P” from “4” to capture the entire share of the market to increase profit. So, it is totally clear that “P > 3” is not a NE.

c).

Now, let’s assume that they charging same price P=3, firm-1 have MC1=1 and firm2 have MC2=2. So, at this price total production will be “Q = 9 – 3 =6”. Now, since they are producing same product and also charging same price, => they will supply equal amount to the market that is Q1=Q2=3. Now, since P=3 > MC1, MC2, => both the firms are earning positive profits.

Now, here also each firm have an incentive to charge less “P” compare to its rival firm, to capture the entire market, and to earn more profit. So, it is totally clear that both of them will think and act in a same way => both of them will reduce “P” from “3” to capture the entire share of the market to increase profit. So, it is totally clear that “P = 3” is not a NE.

d).

As, we know that MC1=1 and MC2=2, => if one firm is charging P=3, so the other firm will not charge more than that “3”. Since, if any one will do that then that firm will not able to capture any portion of the market. Now, on the other hand if one firm is charging P=3 and the other one will charge less than 3, then the latter one will capture the entire market, so the former firm will also reduce price since here both of them have MC < 3.

So, this is also not a NE.

e).

Now, let’s assume that they charging same price P=2, firm-1 have MC1=1 and firm2 have MC2=2. So, at this price total production will be “Q = 9 – 2 =7”. Now, since they are producing same product and also charging same price, => they will supply equal amount to the market that is Q1=Q2=3.5. Now, since P=2 > MC1and equal to MC2, => firm-1 are getting positive profit and firm-2 are getting “0” profit.

Now, here firm-1 have an incentive to charge less “P” compare to its rival firm-2, to capture the entire market, and to earn more profit, on the other hand firm-2 can’t reduce price further.

So, “firm-1” will reduce price and ‘firm-2” will keep charging same price, => firm-1 will be able to capture the entire market. So, it is totally clear that “P = 2” is also not a NE.

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