1. Lee's utility function over goods 1 and 2 is given by:

u (x₁, x₂) = 4√x₁x₂

Lee is currently located in a village where his income is y = $100. There is no other source of wealth. The price of good 1 in the village is always p_i = 1. The supply of good 2 in the village is however rather erratic. Half the time, its the supply is high; at such times its price is p₂ = 1. Another half of the time, its supply is low, and consequently its price at such times is p2 = 4. Before today's market price (which could be p2 = 1 with probability 1/2, or p2 = 4 with probability 1/2) is announced, a retailer comes along offering to sell good 2 at a fixed price of i3

(a) Find Lee's Marshallian demands for goods 1 and 2, and his indirect utility function.

(b) What is the highest price 15 at which Lee will buy from this retailer rather than wait to purchase in the market?

2. Sam resides in a town where the prices of the two goods are p₁ = p₂ = 2. Thus, his utility function is

u(x₁, x₂) = x₁^1/4x₂^1/4

He currently has 0 wealth and is contemplating between two commission-based job offers, both with uncertain salary. Offer 1 is for a job where he may earn either 8100 with probability 1/2 or earn 8900 with probability 1/2 . Offer 2 is for a job which pays him either $64 with probability 1/4 or S with probability 3/4.

(a) Show that Sam's indiret utility function is v (m) = √m/2 given p₁ = p2 = 2.

(b) How high must S be for Sam to accept offer 2?

3. Consider the following 5 x 3 game between Player 1 (row) and Player 2 (column). The first element in any square represents the payoff of Player 1 and the second element, the payoff to Player 2.

(a) Specify the strategies for Player 1 and Player 2, respectively. See if you can find any strategies for Player 1 that is strictly dominated by another; do the same for Player 2. Eliminate these strictly dominated strategies, and con­sider the reduced game. Repeat this same step of deletion of strictly dominated strategies in the reduced game. Continue this process until you can reduce the game no further, i.e., no player has a strictly dominated strategy.

(b) At this point, you should have arrived at a 2 x 2 game. Find all the Nash equilibria for this game, including mixed-strategy ones.

4. Consider the following strategic game between Player 1 and Player 2. Both players move simultaneously. Each has two possible strategies: Help or Fight. The payoffs for this game are given below:

(a) Find all Nash equilibria of this game, including the mixed-strategy ones. Explain why each is an equilibrium.

(b) Suppose instead of making their moves simultaneously, Player 1 makes his/her move first (i.e., to choose Help or Fight). Then Player 2 makes his/her move, i.e., to choose Help or Fight, after observing the choice of Player 1. The payoffs are the same as given by the above table. Draw the game tree of this sequential game.

5. Consider the Bertrand oligopoly model we studied in class. Define P_min = min [P_1,.........p_n]  , Prove the following:

(a) If P_min = c and only one firm is charging the price c, then the price vector [P_1,.........p_n] is not a Nash Equilibrium.

(b) If A_nn > c, then the price vector (p₁, ,p_n) is not a Nash Equilibium.