Intermediate Microeconomics Problem Set 5: A Gauntlet of Economics Part 1: General Equilibrium Two consumers have the following preferences for goods x and y: 𝑢1 = 𝑥1 1/3 𝑦1 2/3 𝑢2 = 𝑥2 2/3 𝑦2 1/3 There are 100 units of x available, and 100 units of y available.

Intermediate Microeconomics Problem Set 5: A Gauntlet of Economics Part 1: General Equilibrium Two consumers have the following preferences for goods x and y: 𝑢1 = 𝑥1 1/3 𝑦1 2/3 𝑢2 = 𝑥2 2/3 𝑦2 1/3 There are 100 units of x available, and 100 units of y available. Both consumers start with equal quantities of the two goods. Using this information, answer the following questions: 1. Depict the starting point for this exchange using the Edgeworth both. Include as much detail as you can, such as labeling indifference curves, initial endowments, and regions of mutually beneficial trade. 2. Define the contract curve between these two consumers and explain what it means. Accurately graph the contract curve on your graph from question 1. 3. Determine the core1 for these two consumers given their initial endowment. Mark this area down on your graph. 4. Calculate the price ratio between x and y. 5. Given this ratio of prices, where do you expect the two consumer’s to end up? Part 2: Game Theory Below is a game: Player B A B C Player A A 10, 5 18, 9 8, 2 B 6, 12 1, 13 7, 6 C 8, 8 7, 9 2, 3 6. Identify all dominant strategies for the players. 7. Identify all Pareto optimal outcomes. 8. Identify all pure strategy Nash equilibria. 9. Identify all mixed strategy Nash equilibria. 1 You are going to want the help of a machine for this one. Here is a link to WolframAlpha with a query similar to the one you’ll need to type in: https://www.wolframalpha.com/input/?i=solve+1%2Fx+%3D+x%2F%281%2Bx%29+for+x. Part 3: Intertemporal Decisions Suppose that the representative consumer has preferences given by the following utility: 𝑢𝑡 = ln 𝑐𝑡 + ln 𝑙𝑡 Which is that utility at time t is equal to the natural log of consumption at time t plus the natural log of leisure at time t. For the sake of mathematics, I am going to “normalize” time so that 24 hours is represented by 1. Or, in other words, the values for leisure and time spent working represent the fraction of the day spent doing those things, rather than the number of hours. The wage will therefore have a similar interpretation. The budget constrain for this consumer is thus: 𝑤𝑡 = 𝑐𝑡 + 𝑤𝑡 𝑙𝑡 + 𝑠𝑡 Assume that consumers have both a discount rate and face an interest rate on savings. For mathematical ease, let these be represented by the following: 𝑝 = 1 1 + 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑟𝑎𝑡𝑒 & 𝑟 = 1 + 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 10. Get expressions for 𝑐1 and 𝑙1 that are functions of only the wage assuming that this model has only one period (so 𝑠𝑡 = 0, 𝑝 = 0, 𝑟 = 1). 11. Get expressions for 𝑐1, 𝑙1, 𝑐2, and 𝑙2 that are functions of only the wages, p, and r. 12. Under what conditions will people borrow money? Present this as a mathematical expression and explain what it means in plain English. 13. If people retire in the second period (meaning their wage falls to 0), what happens to their choices of consumption and leisure in each period? Defend your answer with mathematics.

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