Question

Price Matching

Bernie and Manny both sell DVD players, and both have a per-unit cost of $250 (and no capacity constraint or any fixed costs). They compete on price: the low price seller gets all the market, and Bernie and Manny split the market equally if they have equal prices. The monopoly price for DVD players (the price that maximizes sum of the profits of both firms) is $300.

a. Explain why the only Nash equilibrium has both Bernie and Manny charging $250, splitting the market, and making zero profit.

Now suppose Bernie advertises that if a customer buys a DVD player from him for $300 and discovers that they can buy it cheaper at Manny’s, Bernie will give the customer a rebate equal to twice the price difference (so, for example, if Manny charges $275, Bernie will give the customer a rebate of ($300−$275)×2 = $50). Suppose Manny advertises a similar policy.

b. Show that it is now a Nash equilibrium for both Bernie and Manny to charge $300, and that both make a positive profit.

c. In light of parts a and b, do you feel you are getting a good price when you buy a product in a store that has a price-matching policy? Make sure that you explain your reasoning. (Remember: Your goal should be to demonstrate what you understand about the game, and not to try and anticipate what answer we would want you to give.)

Answer 1

Answer:-

(a).

The nash equilibrium is that stable state which is each player's best response given the other person's strategy and no player has the incentive to deviate if other person's strategy remains unchanged.

Hence, in this duopoly we will consider the following three cases:

Let p1 be the price that Bernie charges and p2 be the price charged by Manny

Case 1: p1 >p2=cost( $250)

In this scenario, the price charged by Bernie is greater than the price charged by Manny so therefore Manny will capture the whole market and no customer will buy from Bernie so in that case this isn't the best response for Bernie and he has an incentive to deviate by charging a lower price

Case 2: p1<p2=cost ($250)

In this case if price charged by Bernie is lower than the price charged by Manny which is lower than the cost than that wont be feasible as no seller will charge a price below its cost.

Case 3: p1=p2=cost ($250)

In this case if both charge a price equal to the cost than that would be the best strategy for both the firms given other person's response as neither player's has an incentive to deviate from its action because any deviation will lead to losing of market by the firm.

Hence if both the firms charge the same price equal to $250 then that would be the nash equilibrium as none has the incentive to deviate from their action given the response of other firm.

(b).

Hence now if Bernie advertises by giving the rebate equal to twice the price difference if price charged is higher than that of Manny, in that case the net price recieved by Bernie would be $250. If both advertises then each will recieve $250. However if one advertises and other doesnt then one who advertises will get $250 while other gets $275. And if both doesnt advertise then each will get $300.

Hence solving we get nash equilibrium where each diesnt advertises and recieves $300. Hence again nash equilibrium will lead to charging of a price $300 that maximises their profit.

(c).

In this price matching policy, each firm tends to charge a price which is equal to or less than the price charged by its competitor. Hence eventually nash equilibrium would result in the price charged equal to marginal cost faced by each firm. This is Bertrand paradox.