Instructions: Read all questions carefully; answer all questions (each question is worth 20 marks); show all the (nontrivial) steps of your derivations; for your answers, typing is preferred.

QUESTIONS

QI

Consider the following normal-form game:

Left Centre Right

Up 01,0 0,4 2,2

Middle 4,0 1,1 5,5

Down 1,2 2,2 3,3

where a 0 and 0 0.

a) For what values of a and /3 the Nash equilibrium couple of strategies is (Middle, Right)? Explain.

b) For what values of a and 0 the Nash equilibrium couple of strategies is (Up, Left)? Explain.

c) For what values of a and 0 we have that Up and Left are non-rationalizable strategies? Explain.

d) For what values of a and 0 there exists a Nash equilibrium in dominated strategies? Explain.

Q2

Consider the following demand function, p = a — bQ, Q = 7i + 72 + ?3, where 7, denotes firm z’s output, i — 1,2,3; a and b are positive parameters. The three firms choose prices simultaneously and non cooperatively (Bertrand competition). Firm i’s marginal cost of production is constant and equal to Cj 0. Assume that ci — C2 — C3 — c.

a) Derive the best response function of each firm.

Assume now that ci C2 and that ci C3.

b) Find the Bertrand-Nash equilibrium price.

c) Show that, if ci C2 and ci C3, firm 1 makes higher profits than the other firms.

Q3

Consider the following three-player normal form game (Player 3 is the page player):

A Player 3 B

Player 2 A B Player 2 A B

Player 1 A 3, 2,1 1,2,3 Player 1 A 1,2,3 2,1,3

B 1,3,2 3,1,2 B 2,3,1 3,2,1

For each player, strategies are A and B.

a) Find the pure strategy Nash equilibrium of the above game.

Assume now that the above game is played sequentially, with Player 3 moving first, Player 1 moving second, and Player 2 moving third.

b) Write down the game tree.

c) For each player, list all the available pure strategies.

d) Find the subgame perfect equilibrium outcome and compare it with the equilibrium outcome you found in subquestion a). Comment.

Q4

Consider the following demand function, p = 1 — Q, Q = J2i=i qt, where qi denotes firm z’s output. Assume that the total cost of firm 1 is q^/2 and the total cost of firm 2 is cq^, with c 6 (0,2/3). Assume that firm 2 chooses its quantity ahead of firm 1 (firm 2 is the Stackelberg leader).

a) Derive the Stackelberg equilibrium (price, quantities, and profits).

b) Is there a first-mover advantage in this game? Explain.

Q5

Two firms are considering whether and how to enter a new industry. Industry demand is given by p = 900 — qi — qz, where p is the market price, q± is the quantity produced by firm 1, and 72 is the quantity produced by firm 2. To enter the industry, a firm must build a production facility. Two types of facility can be built: small and large. A small facility requires an investment of $50,000, and it allows the firm to produce as many as 100 units of the good at zero marginal cost. Alternatively, the firm can pay $175,000 to construct a large facility that will allow the firm to produce any number of units at zero marginal cost. The firms make their entry decisions sequentially. First, firm 1 must choose among staying out of the industry, building a small facility, and building a large facility. Then, after observing firm l‘s action, firm 2 must choose from the same alternatives. If only one of the firms is in the industry, then it selects a quantity and sells its product at the price dictated by the market demand. If both firms are in the industry, then they compete by selecting quantities (as in the Cournot model). All output decisions are subject to capacity constraints

in that a firm with a small production facility cannot produce more than 100 units.

a) Write the above game in extensive form, specifying the correct payoffs the two firms will obtain following their entry decisions.

b) For each player, list all the available pure strategies.

c) Find the subgame perfect equilibrium of the game. How excess capacity can affect entry in an industry? Comment.

QUESTIONS

QI

Consider the following normal-form game:

Left Centre Right

Up 01,0 0,4 2,2

Middle 4,0 1,1 5,5

Down 1,2 2,2 3,3

where a 0 and 0 0.

a) For what values of a and /3 the Nash equilibrium couple of strategies is (Middle, Right)? Explain.

b) For what values of a and 0 the Nash equilibrium couple of strategies is (Up, Left)? Explain.

c) For what values of a and 0 we have that Up and Left are non-rationalizable strategies? Explain.

d) For what values of a and 0 there exists a Nash equilibrium in dominated strategies? Explain.

Q2

Consider the following demand function, p = a — bQ, Q = 7i + 72 + ?3, where 7, denotes firm z’s output, i — 1,2,3; a and b are positive parameters. The three firms choose prices simultaneously and non cooperatively (Bertrand competition). Firm i’s marginal cost of production is constant and equal to Cj 0. Assume that ci — C2 — C3 — c.

a) Derive the best response function of each firm.

Assume now that ci C2 and that ci C3.

b) Find the Bertrand-Nash equilibrium price.

c) Show that, if ci C2 and ci C3, firm 1 makes higher profits than the other firms.

Q3

Consider the following three-player normal form game (Player 3 is the page player):

A Player 3 B

Player 2 A B Player 2 A B

Player 1 A 3, 2,1 1,2,3 Player 1 A 1,2,3 2,1,3

B 1,3,2 3,1,2 B 2,3,1 3,2,1

For each player, strategies are A and B.

a) Find the pure strategy Nash equilibrium of the above game.

Assume now that the above game is played sequentially, with Player 3 moving first, Player 1 moving second, and Player 2 moving third.

b) Write down the game tree.

c) For each player, list all the available pure strategies.

d) Find the subgame perfect equilibrium outcome and compare it with the equilibrium outcome you found in subquestion a). Comment.

Q4

Consider the following demand function, p = 1 — Q, Q = J2i=i qt, where qi denotes firm z’s output. Assume that the total cost of firm 1 is q^/2 and the total cost of firm 2 is cq^, with c 6 (0,2/3). Assume that firm 2 chooses its quantity ahead of firm 1 (firm 2 is the Stackelberg leader).

a) Derive the Stackelberg equilibrium (price, quantities, and profits).

b) Is there a first-mover advantage in this game? Explain.

Q5

Two firms are considering whether and how to enter a new industry. Industry demand is given by p = 900 — qi — qz, where p is the market price, q± is the quantity produced by firm 1, and 72 is the quantity produced by firm 2. To enter the industry, a firm must build a production facility. Two types of facility can be built: small and large. A small facility requires an investment of $50,000, and it allows the firm to produce as many as 100 units of the good at zero marginal cost. Alternatively, the firm can pay $175,000 to construct a large facility that will allow the firm to produce any number of units at zero marginal cost. The firms make their entry decisions sequentially. First, firm 1 must choose among staying out of the industry, building a small facility, and building a large facility. Then, after observing firm l‘s action, firm 2 must choose from the same alternatives. If only one of the firms is in the industry, then it selects a quantity and sells its product at the price dictated by the market demand. If both firms are in the industry, then they compete by selecting quantities (as in the Cournot model). All output decisions are subject to capacity constraints

in that a firm with a small production facility cannot produce more than 100 units.

a) Write the above game in extensive form, specifying the correct payoffs the two firms will obtain following their entry decisions.

b) For each player, list all the available pure strategies.

c) Find the subgame perfect equilibrium of the game. How excess capacity can affect entry in an industry? Comment.

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