Homework 2120 by dxg18808


									Homework #8
Econ 2120
Due 4/24

1.   Consider the city of Apple Valley, Minnesota, where Toys “R” Us and Kmart are the only two major
     competitors in the toy market. It’s approaching the Christmas season, and Talking Elmo Dolls are the
     latest fad. For the purposes of this exercise, assume that the firms have only two possible prices that
     they can charge for Talking Elmos (Low or High). If both firms advertise low prices, they split the
     available customer demand and each earns $2,000 off of sales of these dolls. If both advertise high
     prices, they split the market with lower sales, but their markups end up being large enough to let them
     each earn $3,000. Finally, if they advertise different prices, then the one advertising a high price gets
     no customers and earns nothing while the one advertising a low price earns $4,000. That is, their
     payoff matrix is the following:
                                                       High               Low

                                           High    3000, 3000           0, 4000
                         Toys "R" Us

                                            Low       4000, 0         2000, 2000

     a.   Does Toys “R” Us have a dominant strategy? Does Kmart? What is the Nash equilibrium of this
          pricing game? Is the outcome good for Kmart and Toys “R” Us? Is it good for consumers?

Both Toys “R” Us and Kmart have dominant strategies to choose a low price – whether or not their
opponent is choosing High or Low, it is best for them to choose Low. Thus, the Nash Equilibrium is for
both to choose Low. Despite each choosing their individually best price, the outcome is not good for Kmart
and Toys “R” Us in the sense that they both could be better off if they both chose a High price. It is good
for consumers – lower price means higher consumer surplus.

     b.   Explain how repeated interaction may allow Kmart and Toys “R” Us to collude at the High price.

Repeated interaction, as long as the interaction is infinite or the number of rounds is unknown, may allow
the two firms to collude at the High price. Each firm can condition choosing the High price on whether
their opponent has cooperated and chosen the High price in the past. For example, if Kmart ever chooses a
low price (in order to get 4,000 instead of 3,000), Toys “R” Us might commit to a Low price from then on.
Kmart would thus avoid that outcome because one period of earning an extra 1,000 is not worth an infinite
(or large number) of earning 1,000 less (2,000 instead of 3,000).

     c.   Now suppose that the owner of Toys “R” Us is feeling particularly ruthless this Christmas season,
          and devises a “price matching” policy in an attempt to steal some of Kmart’s market share. The
          matching strategy entails advertising a high price but promising to match any lower advertised
          price by a competitor; Toys “R” Us then benefits from advertising high if Kmart does so also, but
          does not suffer any harm from advertising a high price if the rival advertises a low price. In
          response, the owner of Kmart decides to follow the same policy. Their payoff matrix is thus the
                                                       High               Low

                                           High    3000, 3000         2000, 2000
                         Toys "R" Us

                                            Low    2000, 2000         2000, 2000
         What is the new Nash equilibrium?

The new Nash equilibrium is for both firms to charge a high price. (Side note: technically, both firms
charging a low price is also still an equilibrium, because given the other firm choosing a Low price, there
is no incentive to also not choose a low price.)

    d.   Suppose you are a member of the anti-trust division of the Department of Justice, and are therefore
         interested in consumer welfare. You are investigating the pricing policies of Talking Elmo Dolls.
         What do you conclude about price matching? Does it foster competition?

No, it does not foster competition, but instead allows the firms a mechanism to engage in high price
collusion. It is bad for consumers.

    e.   Suppose that instead of offering either a High or a Low price, each firm can offer ANY price (in
         dollars) for Talking Elmo Dolls. The marginal cost of Talking Elmos is $5. Does either firm have
         a dominant strategy? (Remember, in this case a dominant strategy is a price that is always best to
         charge, no matter what the other firm is charging.) What is the Nash Equilibrium of this pricing
         game, where both firms simultaneously choose a price?

In this case, there is no dominant strategy. If a firm charges $20, for example, the other firm would want to
charge just less than that (say, $19.99) in order to steal all of the sales. If a firm charges $10, the other
firm would want to charge just less than that (say, $9.99). Thus, the optimal price of one firm depends on
the price chosen by the other firm. There is no dominant strategy, since there is no single, always best
price. The Nash Equilibrium of the game will be both firms charging the marginal cost of $5. If the firms
try to charge a price above that, there is incentive for each firm to undercut the other, and to continue
doing so until they get down to marginal cost.

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