questions include equilibrium, game theory, and matrix
1. Consider a retail gasoline market. Assume the following throughout your analysis: (i) a retailer’s variable cost of supplying a gallon of gas is $0.10 plus the wholesale price of the gallon; (ii) retailers’ contracts with wholesalers fix the retail price at $0.25 per gallon above the wholesale price; (iii) annual fixed costs for a retailer equal $50,000; (iv) all wholesalers charge the same wholesale price; (v) all gas stations in the market have equal market share; (vi) gas station owner/managers keep all station profits for themselves; (vii) entry and exit are free, but all gas station owner/managers could earn $100,000 annually in some other job. a. Let Q be the equilibrium amount of gasoline sold in this market in number of gallons, and let N be the equilibrium number of gas stations. What is the mathematical relationship between N and Q? This should be expressed as N = something something Q or Q = something something N. b. Suppose a temporary supply shock, e.g., a hurricane, causes an increase in crude oil prices, thereby increasing wholesale gas prices. Assuming temporary means the short run, what is the impact of the oil price increase on retail gas prices? Show this on a graph. What information would you need to determine the exact effect of this short-run change on the quantity of gas sold and gas station profits? c. What is a shock that would have the same effect on supply but that would make it look more like a permanent change? Assume such a shock happened. Show on your graph what would happen to the demand curve for gasoline. Why would this happen? d. How would this permanent change affect the number of gas stations in the market? Why? 2. Suppose you are the CEO of tech giant Pear, and you have been buying tablet and phone cameras from one supplier, VizTech, for the past two years. Your R&D geniuses at Pear have invented a new technology that will allow the device to use the camera to read the user’s eyes to identify what actions the device should take, without the user having to touch the screen. In order for this to work properly, VizTech would have to make major, expensive modifications to its camera to read minute eye movements and to interface properly with the device. If they didn’t make those modifications, the device would not work, so Pear would continue to use the current camera, and phone and tablet profit would remain the same. If Pear decided not to push forward with their new devices for whatever reason, they would also continue to use the current camera. Your project finance team has calculated that this innovation would increase tablet and phone revenue by $1.5 billion, while tablet and phone costs would increase by $500 million plus the cost of the new cameras. They also calculate that these modifications would add $300 million to VizTech’s costs. Current Pear phone and tablet revenue and cost are $2 billion and $1 billion, respectively. You now have to convince VizTech to modify their cameras according to the specs your team provides them. Assume the specs are 100% accurate and that the technology is guaranteed to work. a. Model this scenario as a sequential game in which VizTech moves first, deciding whether to modify its camera or not. After VizTech makes its decision, Pear has to decide whether to modify its devices, knowing what VizTech decides. Use P as the additional amount that Pear would have to pay VizTech for the new cameras (above what they currently pay). Use a value of 0 for VizTech’s payoff if it does not modify its cameras. Also use a value of 0 for VizTech’s revenue if Pear decided not to modify its devices. Draw your game tree. b. Suppose that Pear and VizTech do not decide on a contractual value of P before VizTech makes their camera modifications. How would this impact the outcome of the game? Use backward induction to justify your response. c. Now suppose that Pear and VizTech are negotiating a contract for P beforehand. What is the minimum value of P that has to be set for the outcome of the game to be that VizTech modifies their camera and Pear modifies their devices? What is the maximum value of P that would result in this outcome? What value of P do you predict? d. Now suppose that VizTech’s success at modifying its camera is not guaranteed. How would this impact contract negotiations, the specific value of P, and the outcome of the game? e. (bonus) Model this game with Pear deciding first whether to modify its devices and then VizTech deciding whether to modify its camera. Does this change the relevant issues or outcome at all? f. Apply your reasoning and analysis from this problem to a situation in your career with which you are familiar. What is the sequence of actions? What are the minimum and maximum transfers at which each outcome would occur? What role does uncertainty play? 3. Suppose that a section of beach facing east has one pier to the north and another pier one mile south of the first one. Two ice cream vendors (N and S) are considering where to place their carts in this one-mile stretch between the piers. Assume that 800 beachgoers are distributed evenly along the beach between the piers. Also assume that all beachgoers like ice cream equally and are indifferent between either of these two vendors. They will simply go to the cheaper vendor, or to the closer vendor if the vendors charge the same price. Assume that these two vendors charge the same price, so their only way to distinguish themselves is their location. Suppose that local ordinances dictate that the vendors can only choose positions at the north pier, ¼ mile from the north pier, ½ mile from the north pier, ¾ mile from the north pier, and at the south pier (1 mile from the north pier). Also suppose that if both vendors choose the same location, vendor N will choose a position that is just barely north of vendor S’s position, meaning that all customers to the north of the two vendors will choose N, and all customers to the south will choose S. Note that this is only true if they choose the same location, as customers always go to the closer of the two vendors. a. Suppose that the vendors choose their locations simultaneously. Then this is a simultaneous move game with the following matrix: ( Vendor N )Vendor S 0 ¼ ½ ¾ 1 0 ¼ ½ ¾ 1 Note that the actions for each vendor are the positions for his or her cart in terms of distance from the north pier. The payoffs of this game are the numbers of customers (out of the 800 total) that each vendor gets from the corresponding combination of positions. Draw a picture of the beach representing this game. Fill in the matrix with the appropriate payoffs and find the Nash Equilibria. b. Now assume that instead of a completely uniform distribution of beachgoers across the entire section of beach, 600 beachgoers are distributed evenly along the northern half of this section, and 200 beachgoers are distributed evenly along the southern half. All other assumptions remain the same. Draw a new picture representing this different version of the game. Fill in the matrix below with the new payoffs and find the Nash Equilibria. ( Vendor N )Vendor S 0 ¼ ½ ¾ 1 0 ¼ ½ ¾ 1 c. How does this model provide insight into the location choices of CVS and Walgreens, gas stations, or restaurants? d. How does this model provide insight into political campaign positioning, particularly moving between party and general elections?