Homework 5-
1. Consider a political campaign where Bob Smith is running against Jane Brown for office. Both candidates are considering whether or not they should engage in negative campaigning: they know that negative campaigning will harm their reputations, but they also both realize that a failure to campaign negatively when their opponent campaigns negatively will result in their losing the political contest. So, for Jane Brown she can either campaign positively and have a tight race with Smith if Smith stays positive; or campaign negatively and win the race if Smith stays positive; or campaign negatively and have a tight race if Smith goes negative; or campaign positively and lose the race if Smith campaigns negatively. Campaigning positively will give her a good reputation while campaigning negatively will give her a bad reputation. Smith faces the same choices and the same payoff. Both Smith and Brown care more about winning the election than they do about any damage to their own reputations.
a. Using a payoff matrix provide a visual representation of the choices these two candidates face given the above information. Make sure your payoff matrix provides information about the political contest as well as the reputation of the candidates.
b. Identify if Smith has a dominant strategy and if so, what that strategy is. Explain how you got your answer.
c. Identify if Brown has a dominant strategy and if so, what that strategy is. Explain how you got your answer.
d. Given your analysis, what kind of political campaign do you predict Brown and Smith will have? Explain your answer.
2. Consider the children's game of rock/paper/scissors. For those unfamiliar with the game or for those who have not played the game in a long time here are the basics of the game. The game is played by two individuals: for example, Robin and Susie. Both players count to three and simultaneously either form one of their hands into a fist symbolizing the rock, form one of their hands into a fist with the first two fingers extended to symbolize scissors, or form on of their hands into a flattened, but open hand, to symbolize paper. The winner of the game is determined as follows: rock beats scissors, scissors beat paper, and paper beats rock.
a. Given the above information construct a payoff matrix for this game.
b. Given the matrix you constructed in (a), does Robin have a simple, pure dominant strategy? Explain your answer.
c. Given the matrix you constructed in (a), does Susie have a dominant strategy? Explain your answer.
d. What do you predict will be the outcome of the game? Explain your answer.
3. Consider a market that has two identical firms. Suppose these two firms decide to form a cartel and select the profit maximizing quantity for the cartel and the profit maximizing price for the cartel (essentially the two firms decide to act as a single monopoly firm). The firms know that the market demand for their product is given by the equation P = 2000 - 1.5Q and the TC for production of this good is given by TC = 100,000 + (1/2)Q2. From this information, the firms deduce that the MC for their joint production is MC = Q.
a. What price and quantity will these two firms decide to produce if they adhere to the cartel agreement? What will be the value of joint profits when the two firms cooperate? If these two firms cooperate and then split the joint profits evenly, what will be the level of profits earned by each firm? Show how you calculated your answers.
b. Suppose that one of the firms decides to cheat on the cartel agreement and sell a total of 600 units of the good at the single price people are willing to pay given the market demand curve for the product. Effectively this cheating firm captures the entire market when it does this (the other firm sells none of the product). Calculate the cheating firm's profit when it deviates from the cartel agreement. Calculate the non-cheating firm's profit under this scenario. Show your work.
c. Now, construct a payoff matrix showing the profits for the two firms given the two strategies (adhere to agreement, or cheat on agreement) that the two firms can pursue.
d. Analyze the payoff matrix you constructed in (c): does each firm have a dominant strategy? If so, identify the dominant strategy. What do you predict will be the outcome of this particular game? Explain your reasoning.
4. Consider the market for fossil fuels. Let's suppose that the market demand for fossil fuels is given by the equation P = 50 - Q where P is the price per unit and Q is the quantity of units of fossil fuels. Furthermore, assume that this market demand curve includes not only the marginal private benefits from consuming fossil fuels, but also the marginal social benefits from consuming fossil fuels. That is, there is no externality problem on the demand side of this market (this is a simplifying assumption for the work we are going to do). On the supply side of the market we know the following: the marginal private cost (MPC) of producing fossil fuels is given by the equation MPC = 10 + Q and are also told that for each unit of fossil fuels supplied that there is an externality cost of $10 per unit.
a. Given the above information, is this a situation of a positive externality or a negative externality? Explain your answer.
b. Given your answer in (a), do you predict this market will produce too much fossil fuel or too little fossil fuel from the perspective of what is the socially optimal amount of fossil fuel? Explain your answer.
c. Given the above information, what will be the market equilibrium quantity and price of fossil fuel if there is no intervention in this market to insure that the externality is accounted for? Show how you got your answer.
d. Given the above information, provide an equation for the marginal social cost (MSC) of producing fossil fuel.
e. Calculate the quantity of fossil fuel that is the socially optimal amount of fossil fuel as well as the socially optimal price per unit of fossil fuel given the above information. Show your work.
5. Imagine a community that has only two residents: Paul and Sally. Paul and Sally both realize that their community would benefit from the installation of streetlights, and they are trying to figure out the optimal number of streetlights for their community. Paul and Sally are both willing to reveal their preferences for streetlights:
Paul's preferences for streetlights are given by the equation: MPB = 20 - 2Q
Sally's preferences for streetlights are given by the equation: MPB = 10 - (1/2)Q
where MPB is the marginal private benefit for the individual and Q is the number of streetlights. Both Paul and Sally know that the marginal social cost of installing a streetlight is given by the equation:
Cost of installing streetlights: MSC = 3.5Q
a. Given this information, start by constructing three graphs that are vertically drawn one under the other. In the top graph depict the MPB for Paul; in the second stacked graph depict the MPB for Sally; and in the final stacked graph depict the MSB (marginal social benefit) from streetlights assuming that Paul and Sally are the only two individuals in this community. Label all axis, and any "kink" points clearly and completely.
b. Given your graphs in (a), write an equation(s) for the MSB curve you found. Make sure you also indicate any relevant range for the MSB equation(s) you provide.
c. Calculate the optimal number of streetlights in this community and then determine the total price per streetlight and the amount of this price that will be paid by Paul and the amount that will be paid by Sally. Show your work and explain how you found your answers.