Download Practice Problems II

Practice Problems III
1. Consider the game of rock-paper-scissors. There are two players, and each of them
chooses to play R (rock), P (paper), or S (scissors). R beats S, S beats P, and P beats R.
When one player’s choice beats the other player’s choice, the winner must pay the loser
$1.00. When both make the same choice, it’s a tie, and no money changes hands. Draw
a payoff matrix representing this game. Then identify any Nash Equilibria of the game.
2. Suppose Akeem and Rojelio both have access to a magic hat. There is $100 in the hat.
Each day at noon, Akeem and Rojelio each have the option of reaching into the hat and
grabbing as much money as possible. If only one reaches in, he gets all the money in the
hat. If both reach in, each gets half the money. Any money left in the hat overnight will
increase by 50%. To simplify the game, suppose that there only two days, today and
tomorrow. Since both players will definitely reach in the hat tomorrow, the only question
is whether to reach in today. Draw a payoff matrix representing this game. Then identify
any Nash Equilibria of the game. What kind of game is this?
3. Hector and Menelaus entered into a business contract. It seemed like a good idea at
the time, but now Hector realizes he wants out. But according to the law in their country,
contracts like this must be fulfilled unless both parties agree to void the contract (that is,
you can’t just break the contract and pay damages). It’s worth $1000 to Hector to void
the contract, and doing so would make Menelaus $400 worse off than if the contract were
fulfilled. Notice that it makes economic sense to void the contract, since doing so will
create a $600 gain. So Hector offers to buy Menelaus’s consent to void the contract. In
the negotiations, each party must decide whether to drive a “hard bargain” or “easy
bargain.” If Hector drives a hard bargain while Menelaus drives an easy bargain, Hector
will pay Menelaus $500 to void the contract. If Menelaus drives a hard bargain while
Hector drives an easy bargain, Hector will pay Menelaus $900. If both drive an easy
bargain, Hector will pay Menelaus $700. Finally, if both drive a hard bargain, no deal is
made, and the contract must be fulfilled. Draw a payoff matrix representing this game,
and then identify any Nash Equilibria of the game. [Hint: Treat the situation in which
the contract is fulfilled as the baseline, and measure gains relative to that baseline.] What
kind of game is this?
4. (a) Suppose that duopolists engage in Cournot competition. The market demand
curve is P = 140 – 2Q, and each firm has a marginal cost of 20. What are the equilibrium
quantities and price? (b) Now suppose everything is the same, except they engage in
Bertrand competition. Assume the products are homogeneous. What are the equilibrium
prices and quantities? (c) Now suppose everything is the same, except they engage in
Stackelberg competition, with firm 1 committing to its quantity first. What are the
equilibrium quantities and price?
5. Same problem as #4, above, except the demand curve is P = 90 – (1/3)Q, and the
marginal cost is 30.
6. (a) Suppose that duopolists engage in Cournot competition. The market demand
curve is P = 210 – Q, and the firms have different marginal costs: MC = 30 for firm 1,
and MC = 60 for firm 2. What are the equilibrium quantities and prices? (b) Now
suppose everything is the same as in (a), except they engage in Bertrand competition.
Assume the products are homogeneous. What are the equilibrium prices and quantities?
7. Same problem as #6, above, except the demand curve is P = 90 – 2Q, marginal cost is
12 for firm 1, and marginal cost is 18 for firm 2.