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HW12

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Econ 300
Assignment 12

  1. Express the notion of a Nash equilibrium as a utility maximization problem. What does each player maximize? What are the constraints in the problem? What can you say about these constaints at the Nash equilibrium?

  2. Explain why the Cournot equilibrium described in Chapter 11 in the book is a Nash equilibrium in a game where firms' strategies consist of potential output levels. Why is the intersection of firms' reaction functions (Figure 11.3) a graphical illustration of the Nash equilibrium concept?

  3. Player A and B are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, B pays A $1. If they differ, A plays B $1.

    1. Write down the payoff matrix for this game and show that it does not contain a Nash equilibrium.

    2. How might the players choose their strategies in this case?

  4. Suppose firm A and firm B each operate under the conditions of constant average cost and marginal cost but $ MC_{A}=10$ and $ MC_{B}=8$ . The demand for the firms' output is given by

    $\displaystyle Q_{D}=500-20P.$    

    1. If the firms practice Bertrand competition, what will be the market price under a Nash equilibrium?

    2. What will be the profits for each firm?

    3. Will this equilibrium be Pareto efficient?

  5. The game of ''chicken'' is played by two macho teens who speed towards each other other a single-lane road. The first to veer off is branded the chicken, whereas the one who doesn't turn gains peer group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the chicken game are provided in the following table:


        B's Strategies
        Chicken Not Chicken
    A's Chicken $ 2,2$ $ 1,3$
    Strategies Not Chicken $ 3,1$ $ 0,0$

    1. Does this game have a Nash equilibrium?

    2. Is a threat by either not to chicken out a credible one?

    3. Would the ability of one player to firmly commit to a no-chicken strategy (by, for example, throwing away the steering wheel) be desireable for that player?

  6. The Wave Energy Technology (WET) company has a monopoly on the production of vibratory waterbeds. Demand for these beds is relatively inelastic-at a price of $1000 per bed, 25,000 will be sold; whereas, at a price of $600, 30,000 will be sold. The only costs associated with waterbed prodcution are the initial costs of building a plant. WET has already invested in a plant cabple of productin gup to 25,000 beds and this sunk cost is irrelevant to its pricing decisions.

    1. Suppose a would-be entrant to this industry could always be assured of half the marekt but would have to invest $10 million in a plant. Construct entrant's strategies (enter, don't enter). Does this game have a Nash Equilibrium?

    2. Suppose WET could invest $5 million in enlarging its existing plant to produce 40,000 beds. Would this strategy be a profitable way to deter entry by its rival?

  7. The following table reports the payoff matrix for an advertising game. Explain why the strategy pair ''A: high, B: low'' is a Nash equilibrium in this game and all the other strategy pairs are not.
        B's Strategies
        High Low
    A's High $ 5,2$ $ 3,3$
    Strategies Low $ 4,3$ $ 2,4$

  8. Two firms (A and B) are considering biring out competing brands fo a health cigarette. Payoffs to the comparnies are as follows (A's profits are given first):
        B's Strategies
        Produce Don't Produce
    A's Produce $ 3,3$ $ 5,4$
    Strategies Don't Produce $ 4,5$ $ 2,2$

    1. Does this game have a Nash equilibrium?

    2. Does this game present any first-mover advantages for either firm A or firm B?

    3. Would firm B find it in its interest to bribe firm B to say out of the market?


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Jenn Thacher 2008-08-25
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