Question 5: 120 points] Consider two firms selling an identical product who comp
ID: 1165236 • Letter: Q
Question
Question 5: 120 points] Consider two firms selling an identical product who compete in price (Bertrand competition with homogeneous products). The marginal cost of each firm is 10 and inverse market demand for the product is P = 50-Q. Assume Firm I can only set even integer prices and Firm 2 can only set odd integer prices. (a) [10 points] ls (P1,P2)-(14, 13) a Nash equilibrium? Justify your answer. (b) [10 points] Find the Nash equilibrium or Nash equilibria. You must show why what you have found is/are a Nash equilibrium. Points will be deducted if you include pairs of prices which are not Nash equilibria.Explanation / Answer
(a) Nash Equilibrium is a set of strategy/ strategies for each player where in no player has the incentive to deviate from or change his or her strategy given what the other player is doing.
Here, (p1,p2) = (14,13) is not a Nash equilibrium because at this price the whole market will buy from firm 2 and firm 1 will earn 0 profit therefore, firm 1 has the incentive to change its price and reduce it so that it can capture the whole market and earn larger profit.
Firm 1 can reduce its price to 12(even integer price) which is lower than the price of firm 2, there by capturing the whole market(as its price is lower than that of product of firm 2, therefore whole market will buy the product from firm 1) and this price is more than its marginal cost of 10 which makes it profitable for firm 1.
Hence, (p1,p2) = (14,13) is not Nash equilibrium.
(b) We have,
P = 50-Q
this implies, Q = 50-P
Let p1 be the price set by firm 1 and p2 be the price set by firm 2, then,
Profit for firm 1= (P1-MC)Q
= (P1-MC) (50-P1) {because Q = 50-P}
= (P1-10) (50-P1)
=50P1-P12-500+10P1
First order differentiation condition gives,
P1=30
Since, demand function and marginal cost for both the firms is same, therefore, by symmetry, profit function of firm 2 will given as (P2-MC)Q
= (P2-MC) (50-P2) {because Q = 50-P}
= (P2-10) (50-P2)
=50P2P22-500+10P2
First order differentiation condition gives,
P2=30
Therefore, profit of both the firms is maximum when each of them sets the price equal to 30 but firm 2 cannot set a even integer price, so the best option for firm 2 is to set price equal to 29 so that it can maximise its profit.
But, (p1,p2) = (30,29) is a not Nash equilibrium because at this price whole market will buy from firm 2 as its price is lower than that of firm 1, therefore, firm 1 has the incentive to reduce its price to 28(even integer price just lower than 29) so that it can serve the whole market and earn larger profit.
This strategy of reducing price will go on until they reach the marginal cost.
At (p1,p2) = (12,11),
firm 1 is indifferent between setting price equal to 12 and 10 since it earns 0 profit at any of these prices, because if it sets price equal to 12 and firm 2 sets its price equal to 11 then whole market will buy from firm 2. And if it sets its price equal to 10 i.e. p1= marginal cost then also profit equals 0 for firm 1.
Therefore, (p1,p2) = (12,11) is a Nash equilibrium since no firm has the incentive to change its price in this situation.
Similarly, (p1,p2) = (X,11), where X is any even integer more than equal to 10 is a Nash equilibrium since firm 1 is indifferent between setting price equal to 10 i.e. its marginal cost or X and it will not want to set its price less than 10 because it will lead to losses for firm 1.
Therefore, (p1,p2) = (X,11) is a Nash equilibrium since no firm has the incentive to change its price in this situation where X is any even integer more than equal to 10.
At (p1,p2) = (10,11), where whole market will buy from firm 1 as it has a lower price, firm 2 has no incentive to set its price equal to 9(odd integer price just lower than 10) and serve the whole market as this price is lower than marginal cost of 10 and will lead to losses for firm 2.
Therefore, (p1,p2) = (10,11) is a Nash equilibrium since no firm has the incentive to change its price in this situation.
Similarly, (p1,p2) = (10,Y), where Y is any odd integer more than equal to 11 is a Nash equilibrium since firm 2 is indifferent between setting price equal to 10 i.e. its marginal cost or Y(0 profit in either cases) and it will not want to set its price less than 10 because it will lead to losses for firm 2.
Therefore, (p1,p2) = (10,Y) is a Nash equilibrium since no firm has the incentive to change its price in this situation where Y is any odd integer more than equal to 11.
At any price lower than 10, firms will incur losses, therefore, no firm will set a price lower than 10.
Hence, Nash equilibria are: (p1,p2) = (X,11) where X is any positive integer more than equal to 10 ?and (p1,p2) = (10,Y) where Y is any odd integer more than equal to 11
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