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1. Sarah consumes only two goods, which can be divided into fractions: ap ples (:31) and books (.132). Assume the price of books is p2 = 1. Suppose you know that, if she consumes 5 books, she can buy at most 10 apples. And if she consumes 8 books, she can buy at most 4 apples. (3) (b) (C) (d) Determine her income (m) and the price of apples (p1). Write out the equation of her budget line. Draw this on a graph (clearly marking the axes and intercepts) and indicate her budget set. Suppose the government taxes Sarahls apple consumption in the fol- lowing way: for each apple consumed, Sarah must pay a tax of t = %. Write out the equation for her new budget line, and draw it on a graph. After evaluating this policy, the government decides that the tax is too harsh. It now declares that the tax will kick in only after 5 apples. (If 1:1 <_c there is no tax. but if> 5, the tax applies to each apple that exceeds this cutoff.) Sarah will now face a kinked budget line. Write the equations corresponding to each part of her budget line, and draw them on a graph (indicate the bundle at which the kink occurs). This government is extremely indecisive. In another change of policy, it decides that the tax break makes sense for the poor, but there is no reason to oer it to the rich. The new policy is as follows: if a person's income is less than or equal to $10, she gets the tax break (as described in part c), but if her income is greater than $10, she must pay taxes on her entire purchase of apples (as described in part b). Now suppose Sarah's employer offers her a raise of $2. Can we be certain that Sarah will accept the raise? What about a raise of $4? Analyze this by thinking about how the budget set changes after each ralse. This describes a strategic game with ordinal preferences. Twe testes of volun- teers are taking part in a competition to solve a math problem. A reward will be given to the team that solves the problem rst. The volunteers on a team work in parallel. If one volunteer succeeds in solving the math prohlem.1 they tell the team leader who shares the reward equally among the volunteers. Each team could send an undercover spy to the other team (the host team). The spy would work on the math problem but would not tell the host team and thus that team would not collect the reward. However. if another volunteer on the host team solves the problem, the spy would take their share of the reward the host team volunteers get. A few points: or This is a game with two players (each team is a player}. o The actions available to the teams are Spy or No Spy. a a team will get the most income by putting a spy in the other team if at the same time it has no spy on its team. I If both teams use spies. both earn l than if both teams do not. a If a spy is used against a team, the host team will make less money than if both teams used spies. {a} Provide the ranking diagrams [or trees] where the action proles are drawn in levels. {h} Provide the 2 x 2 table using the payoffs from labeling the levels in the ranking diagrams. (c) Are there Nash equilibria for this game? Why or why not? If there are give an explanation why. If not, explain why not. {d} Does this re'semhle any of the games we've studied so far? Provide a complete explanation. 4. Suppose Slava's utility function is U(:1:1,3:2) = (3:1)4 (3:2) and his budget constraint is plwl + p232 = I. (a) Find the formula for an indifference curve that yields some utility c. Graph it. (b) Solve for the MRS at any bundle ($1,232) (using marginal utilities). (c) Now, calculate MRS at any bundle ($1,222) by directly evaluating the slope of an indifference curve and show that your answer is the same as in part (b). (Hint: After taking the derivative of an arbi- trary indifference curve, think about what \"c\" means and substitute appropriately). (d) Briey interpret how the MRS changes along an indifference curve. Assignment 1: Use the knowledge acquired in the course to comment and address the following problem about the relation between deficit reduction and investment. Deficit Reduction: Good or Bad for Investment? You may have heard this argument in some form before: "Private saving goes either toward financing the budget deficit or financing investment. It does not take a genius to conclude that reducing the budget deficit leaves more saving available for investment, so investment increases." This argument sounds simple and convincing. However, in some cases, a deficit reduction may decrease, rather than increase, investment. How can we reconcile such an economic fact with the macroeconomics argument stating that: if the government reduces its deficit - either by increasing taxes or reducing government spending so that t - g goes up - investment must go up? Is the deficit reduction good or bad for Investment? Illustrate your answer verbally and graphically using the goods-market equilibrium condition