Game Theory:

Review Problem:

In a dry dusty desert town, two people (Antoinette and August) have water wells. Each day they begin the morning by pumping water, which they then bring to the town square to sell. Pumping water costs them essentially nothing, but time is such that each can afford to make only one trip to the town center. That means they must decide how much to pump at the start of each day. Due to transportation and container logistics, each can only transport one container of water per day. Each has a limited variety of containers of difference sizes: 20 gallons, 30 gallons, 40 gallons, or 50 gallons. So each has to decide at the beginning of the day which one container they will fill and bring to the town center.

The townfolk’s aggregate market demand for water is as follows:

Price / $12 / $11 / $10 / $9 / $8 / $7 / $6 / $5 / $4 / $3 / $2 / $1 / $0
Q (gallons) / 0 / 10 / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / 100 / 110 / 120
  1. Draw the normal form decision matrix for this game where each of the players can choose different amounts of water to bring to the town center. The payoff matrix will contain the profits each will get depending on which strategies they and the other person employ.
  2. Is there a dominant solution to this problem? (Answer: yes, there is an iterated weak dominant solution) What is it?
  3. Are there Nash equilibria? If so, what are they?
  4. Suppose Antoinette and August agree to collude and restrict their combined output to maximize their joint profits. They agree to divide production and profits evenly between them. How many gallons will they agree to provide in total? Each? What will be the total profits between them? How much will each make?
  5. Is the collusive arrangement in part (d) stable? Or will each have an incentive to cheat if they believe the other will abide by the collusive arrangements?

Budget Lines Indifference Curves and Utility:

Suppose goods x and y both have positive market prices, but good x is also subject to ration coupons that are issued by the government. In other words, if you want to by x, you must both pay the price and provide ration coupons for the amount you wish to purchase. You can buy as much of good y as you want as long as you are willing to pay the price; there are no ration coupons for y. Show how two people can both be made better off if they are allowed to buy or sell the ration coupons from each other. We built the budget lines for this problem in class, so all you need to do is add the indifference curve that show how each person is made better off after the trade.

If Kelly’s utility function is given by U = 5X + 2Y where MUx = 5 and MUy = 2,

  1. Draw indifference curves for U = 10 and U =20. What is the MRSyx?
  2. Can you describe the relationship between X and Y?
  3. If the price of X is 2 and the price of Y is 4, how many of each of these goods will be purchased if m = 30?

In problem 3 of homework set 1,

  1. If indifference curves are “well behaved” i.e., convex to the origin, if the merchant sets a single price of $1.50 for each unit of x, can there be multiple consumer optimums?
  2. If the merchant employs the “quantity discount” pricing scheme, can there be multiple consumer optimums?