Game Theory and Decision Making in Microeconomics

1. Two firms simultaneously determine their output (Q = 50, 100, 150), and then receive profits (in millions of dollars) detailed in the payoff matrix below.

Firm 2
50 100 150
50 6 , 6 4 , 7 2 , 6
Firm 1 100 7 , 4 5 , 5 1 , 3
150 6 , 2 3 , 1 −1 , −1

a. Does any firm have a strictly dominant strategy? How about a strictly dominated strategy?
b. Does this game have any Nash equilibria? If so, what are they?
c. Suppose the two firms choose to collude by forming a cartel. What level of output would the firms decide to sell? If Firm 1 believes Firm 2 will abide by the cartel agreement, what level of output should they set?
2. Consider the sequential game using the same payoffs as above. Firm 1 sets output first, with Firm 2 observing their decision and setting output second.

a. What is the subgame perfect Nash equilibrium of this sequential game?
b. Compare the subgame perfect Nash equilibrium with the Nash equilibrium(ia) in the sequential game. Does Firm 1 prefer to be the leader and move first?

3. The local farmers’ market has (inverse) demand for a box of pumpkin pies P = 330 − Q. Alice and Bob are the only two farmers in the market who sell boxes of pumpkin pie. They both have constant marginal and average cost of $30 per box. Both farmers simultaneously determine the quantity of boxes they are going to bring to market. They do not confer be?fore hand, and the price is determined by the market after they arrive at the farmers market.

a. What are Alice and Bob’s best response functions?
b. What are the Cournot equilibrium quantities and the price that boxes of pumpkin pies sell for at the farmer’s market?
c. Suppose Alice sets her quantity first, and Bob can decide how many boxes to sell only after observing the Alice’s quantity. What is the subgame perfect Stackelberg equilibrium amount of boxes each farmer sells?
d. Does Alice prefer to move first, or would she rather set quantity second after Bob? (you may use any sufficient means to explain this such as a graph or math analysis)
e. Suppose instead that Alice and Bob compete by setting prices instead of quantity, and they do so by setting their prices simultaneously. What will be the new equilibrium price and quantity in the market? What will be Alice and Bob’s profit, and how many boxes of pumpkin pie will each sell?

4. Suppose that Samsung and Amazon are the only manufacturers of e-readers, tablets for reading digital copies of books. The inverse market demand for millions of e-readers in the US is, P = 550 − 10QD.
a. Each firm has constant marginal cost of producing an e-reader of $10. a. What is the Cournot duopoly equilibrium price and quantity in the market. Recall, the Cournot equilibrium is where the firms are competing by setting output (q).
b. What if Samsung and Amazon decided to form a cartel, and fix the price of e-readers to maximize their profits. What is the price of e-readers they would set? How many e-readers would each company sell?
c. Do you expect the agreement to last? Why or why not? Explain specifically for Amazon and Samsung in this example.

5. You are on the TV game show Let’s Make a Deal, hosted by Wayne Brady. You have utility of wealth U(W) = √W.
a. Wayne Brady presents you with a curtain and says it may contain a prize valued at $4,000 or a zonk, which has no value. Both outcomes are equally likely. He asks if you would rather take what is behind the curtain, or $500. Do you choose the curtain or take the money?
b. Wayne Brady increases the offer to $1,000. Do you take the curtain or the money? What if he adds an autographed selfie (which you value at $5) to the $1,000 offer?
c. If you were risk loving, and had utility of wealth U(W) = W2 , would there be a sum of money below $4,000 you would accept to not take the gamble?

6. Ron has $10,000. Ron has a one-in-five chance of developing an illness that will cost $7,500 to treat if he has it. Ron’s utility of wealth is U(W) = √W
a. What is Ron’s expected utility if he does not know if he has the illness?
b. Suppose Ron can purchase full insurance that will pay for all of his treatment if he has the illness. What is the actuarially fair premium for this coverage?
c. What is the maximum premium Ron will pay for full coverage?
d. Suppose Ron decides to implement lifestyle changes that reduce his probability of having the illness to one-in-ten. What is the actuarially fair premium for his coverage after the lifestyle change? What is the maximum premium he will pay for full coverage now?